forcesinphysics-historicalperspective

FORCES IN PHYSICS

A Historical Perspective

Steven N. Shore

FORCES IN PHYSICS

A Historical Perspective

Steven N. Shore

Greenwood Guides to Great Ideas in ScienceBrian Baigrie, Series Editor

GREENWOOD PRESSWestport, Connecticut � London

Library of Congress Cataloging-in-Publication Data

Shore, Steven N.Forces in physics : a historical perspective / Steven N. Shore.

p. cm. – (Greenwood guides to great ideas in science, ISSN 1559–5374)Includes bibliographical references and index.ISBN 978–0–313–33303–3 (alk. paper)1. Force and energy—History. 2. Physics—History. I. Title.QC72.S46 2008531′.6—dc22 2008018308

British Library Cataloguing in Publication Data is available.

Copyright C© 2008 by Steven N. Shore

All rights reserved. No portion of this book may bereproduced, by any process or technique, without theexpress written consent of the publisher.

Library of Congress Catalog Card Number: 2008018308ISBN: 978–0–313–33303–3ISSN: 1559–5374

First published in 2008

Greenwood Press, 88 Post Road West, Westport, CT 06881An imprint of Greenwood Publishing Group, Inc.www.greenwood.com

Printed in the United States of America

The paper used in this book complies with thePermanent Paper Standard issued by the NationalInformation Standards Organization (Z39.48–1984).

10 9 8 7 6 5 4 3 2 1

Dedica:a Sephirot

la forza del mio destinotu mi hai insegnato a vivere.

Fabrizio De Andre

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Contents

List of Illustrations ix

Series Foreword xi

Preface xiii

Acknowledgments xv

1 Force in the Ancient and Classical World 12 Medieval Ideas of Force 213 The New Physics 394 From the Heavens to the Earth 655 Forces and Fields 996 Thermodynamics and Statistical Mechanics 1177 Fields as Everything 1378 The Relativity of Motion 1599 Quantum Mechanics 179

Appendix: Some Mathematical Ideas 203

Timeline 215

Bibliography 221

Index 229


List of Illustrations

1.1 A Roman mosiac from the first century AD, discovered at Pompei,depicting the death of Archimedes during the Roman siege andconquest of Syracuse. 9

1.2 The level. Displacement of the left downward moves more mass at alarger distance from the fulcrum (the support point) through thesame angle than the right side (the vertical distance on the left isgreater than that on the right). 10

3.1 The illustration from de Revolutionibus showing the explanation forthe seasons related to the constancy of the direction in space of therotational axis of the Earth during its annual orbit. 40

3.2 A figure from Gilbert’s De Magnete showing the “influence” of themagnetism of a lodestone in the form of a sphere, the dip of amagnetic needle. This model of the Earth’s magnetic field, theterrela, provided the empirical basis for later attempts in theseventeenth century (notably by Edmund Halley) to determinelongitude with magnetic measurements. 44

3.3 Galileo. 483.4 An illustration from Galileo’s Two New Sciences showing the

scaling of a bone as an illustration of the application of statics andbreaking tension to the size of animals. 50

3.5 The discovery of the presence and motion of the satellites of Jupiterfrom Galileo’s Sidereus Nuncius. This was the first demonstration ofbound orbital motion around another planet and showed thereasonableness of the Copernican construction for a heliocentricplanetary system. 52

3.6 Rene Descartes. 593.7 Figure from Descartes’ Principia showing his explanation for

centrifugal force. A body tends to recede from the center of a curve,

x List of Illustrations

the sling maintains the motion around the curve against this. Whilethis is true in the moving frame, Newton demonstrated that theunconstrained motion is really along the instantaneous tangent tothe curve in a straight line, not radially outward. 60

4.1 The Newtonian construction of centripetal force. A particle movingalong a circular path centered at any moment at some distance Rhas a tangential velocity V. The direction of this motion changesalthough the speed remains constant. The acceleration, thedirection of the change in the tangential component, is toward thecenter. The force is, therefore, in the same direction according tothe second law and equal in magnitude and opposite to the changein the motion by the third. 69

4.2 Newton’s example for orbital motion from the Principia. Aprojectile is launched with increasing initial horizontal speeds.Eventually there is a critical value at which the body falls at thesame rate as it displaces horizontally, maintaining a constantdistance. Thus circular motion is always accelerated – it’s alwaysfalling but never hitting the ground. 72

4.3 Figure from Descartes’ Principia showing his explanation forgravitation based on a universal fluid filled with contiguousvortices. Each star is the center of a vortex. This is the world laterdescribed by Voltaire and contrasted with the Newtonian vacuum.It was this model for planetary motion that spurred Newton’scomposition of the second book of his Principia. 76

4.4 Isaac Newton. 774.5 Leonhard Euler. 884.6 The bending of a beam showing the differential strain and stress

described by the two-dimensional application of Hooke’s law ofelasticity. If the weight is released, the beam will oscillate aboveand below the horizontal. 89

5.1 The construction of gradients through equipotential surfaces, anillustration of Green’s idea of the potential. 104

5.2 In the background of this portrait of Nevil Maskelyne, notice themountain that was the object of his study of the gravitationalattraction of the Earth. 106

5.3 Joseph Louis Lagrange. 1107.1 Hermann von Helmholtz. 1457.2 Michael Faraday. 1467.3 James Clerk Maxwell. 1498.1 Albert Einstein. 1668.2 A cartoon comparison of the classical (Newtonian) and relativistic

descriptions of the deflection of a body by another mass. In the leftpanel, the trajectory is changed because of a force. In the right, onecan describe the motion along a curved spacetime (here shownschematically as a potential well but the motion is really in fourdimensions, the time changes along the trajectory). 166

9.1 Niels Bohr. 185

SERIES FOREWORD

The volumes in this series are devoted to concepts that are fundamental to differ-ent branches of the natural sciences—the gene, the quantum, geological cycles,planetary motion, evolution, the cosmos, and forces in nature, to name just a few.Although these volumes focus on the historical development of scientific ideas,the underlying hope of this series is that the reader will gain a deeper under-standing of the process and spirit of scientific practice. In particular, in an age inwhich students and the public have been caught up in debates about controversialscientific ideas, it is hoped that readers of these volumes will better appreciatethe provisional character of scientific truths by discovering the manner in whichthese truths were established.

The history of science as a distinctive field of inquiry can be traced to the earlyseventeenth century when scientists began to compose histories of their own fields.As early as 1601, the astronomer and mathematician Johannes Kepler composeda rich account of the use of hypotheses in astronomy. During the ensuing threecenturies, these histories were increasingly integrated into elementary textbooks,the chief purpose of which was to pinpoint the dates of discoveries as a way ofstamping out all too frequent propriety disputes, and to highlight the errors ofpredecessors and contemporaries. Indeed, historical introductions in scientifictextbooks continued to be common well into the twentieth century. Scientistsalso increasingly wrote histories of their disciplines—separate from those thatappeared in textbooks—to explain to a broad popular audience the basic conceptsof their science.

The history of science remained under the auspices of scientists until theestablishment of the field as a distinct professional activity in the middle of thetwentieth century. As academic historians assumed control of history of sciencewriting, they expended enormous energies in the attempt to forge a distinct andautonomous discipline. The result of this struggle to position the history of scienceas an intellectual endeavor that was valuable in its own right, and not merely

xii Series Foreword

in consequence of its ties to science, was that historical studies of the naturalsciences were no longer composed with an eye toward educating a wide audiencethat included nonscientists, but instead were composed with the aim of beingconsumed by other professional historians of science. And as historical breadthwas sacrificed for technical detail, the literature became increasingly dauntingin its technical detail. While this scholarly work increased our understandingof the nature of science, the technical demands imposed on the reader had theunfortunate consequence of leaving behind the general reader.

As Series Editor, my ambition for these volumes is that they will combinethe best of these two types of writing about the history of science. In step withthe general introductions that we associate with historical writing by scientists,the purpose of these volumes is educational—they have been authored with theaim of making these concepts accessible to students—high school, college, anduniversity—and to the general public. However, the scholars who have writtenthese volumes are not only able to impart genuine enthusiasm for the sciencediscussed in the volumes of this series, they can use the research and analyticskills that are the staples of any professional historian and philosopher of scienceto trace the development of these fundamental concepts. My hope is that a readerof these volumes will share some of the excitement of these scholars—for bothscience, and its history.

Brian BaigrieUniversity of Toronto

Series Editor

PREFACE

The history of physics is the conceptual history of force. It is also the story ofhow quantitative analysis became ever more central to, and effective for, exploringphysical phenomena. Thus, a few basic questions will flow as a theme throughoutthis book: “What is motion?”, “Why is there motion?”, and “How is there motion?”These are so elementary that they may strike you as trivial. After all, you knowhow to describe the motion of a ball rolling along a bowling alley or flying througha stadium. You know that it is moving (it’s changing its location relative to youand its target), why it is moving (it was thrown or struck or kicked by someone,probably intentionally), and how it is moving (it has some speed and goes in somedirection). But woven through the tapestry of the argument in this whole book willbe an effort to make such commonplace observations terribly difficult becausethose who struggled with them for so many centuries thought they were and that’sthe only way to step into that now vanished world. In these amazingly ordinaryquestions you find the origin of physics, the effort to understand the laws governingthe structure of the material world that spans more than two millennia from thetime of the Greeks to the present.

The order of presentation is historical, beginning in antiquity, but don’t mistakechronology for linearity. The difficulty in discussing the conceptual developmentof science is how it proceeds simultaneously along many lines of inquiry. Some-times random coincidences strike a resonant chord and produce new, unexpected,results. Philosophers of science have puzzled over this for generations. These maycome from the accumulation of paradoxes, problems, and contradictions during aperiod between great leaps, or they may arise from simple “what if” questions.The history of force displays both modes of discovery. But science in general, andphysics in particular, is the product of communication among those who engage inthis grand activity. It is a collective activity with a rich history. If we think thereare really laws of nature then, in one sense, it doesn’t actually matter who finds

xiv PREFACE

them—eventually they’ll be discovered. For a working physicist, attempting to un-derstand and wring something fundamental out of experiments and observations,theory is about how to best represent these laws and the phenomena they produce.It is about models, how we approximate reality. That’s easily stated but I rememberreading this sort of statement when I began studying science and it made no senseat all. Now, I think, it’s because the presentation was often completely ahistorical.The laws seemed to come out of nowhere. The same thing usually happened inmathematics. Worse still was the presentation of the “scientific method,” whichseemed almost silly. Why would you ever think of testing something, just as anidea out of the blue? How are the hypotheses formed in the first place and why? Ihope you’ll find some explanation of that in this book.

I should explain what I won’t be doing here. There are several other books inthis series that deal with the development of astronomical, cosmological, chemical,and biological ideas. You should approach this work keeping in mind its broadersetting. While I discuss some of the developments in these areas, the presentation isintended to be contextualizing rather than complete. Some parts will seem sketchy,especially when describing planetary theory from the time of the Greeks throughthe later Middle Ages. That’s intentional and would seem to require a separatebook, as indeed it does in this series. It’s difficult to write a purely descriptivehistory of any physical science, but this is particularly true for mechanics. Itwas the font of much of modern mathematics, and the symbiosis that marks itsdevelopment forms an important part of the story. A word to the reader is in orderregarding the subsequent increasingly mathematicized discussion of this book.At the end of the seventeenth century, a new method was introduced into physics,the infinitesimal calculus or, as it is now called, differential and integral calculus.The mathematical advance was to recognize that the limit of finite impulses, forinstance, could produce in an instant of time a finite acceleration or displacementand that it is possible to describe a continuous variation in any quantity. Formotion and force this can be regarded as the most important step in the entirehistory of physics. Equipped with this new tool, Newton’s law can be written in acontinuous, invariant form that redefines acceleration and renders impulse a limitof a continuously acting force. One-dimensional histories can be expanded tothree, and changes explicitly relative to time and space can be separated. Further,the history of a motion—a trajectory—can be described with the continuous sumcalled an integral. This is just an example, there are others and I’ve added a shortappendix that gives a few more details. I’ll make no apologies for the increasinguse of equations and symbols here. In fact, those who are seeing this for the firsttime will, I hope, find an inspiration for further study in this mathematical physics.Finally, a word about the biographical material and choices of personalities. Manythinkers contributed to the development of physical ideas and it’s not my intentionto make this a set of capsule biographies set within a simple timeline. Instead,while I will focus on a relatively few individuals, the ideas are the principal aim ofthe work and I hope the reader will not find the irregular path traced by the text tobe too confusing. The names and a few key concepts provide the landmarks, butnot the milestones, in our journey. What for me has always been the richness of thestudy of the history of science is how it reveals the recurrence of the fundamentalproblems, often in unexpected and beautiful ways.

ACKNOWLEDGMENTS

I’ve benefitted from advice and criticisms from many colleagues while writing thisbook and it is a pleasure now to publicly repay that debt. I want first to thank BrianBaigrie, the editor of this series, for his kind invitation to participate in the projectand for constant encouragement and precious critiques. Jason Aufdenberg, PierGiorgio Prada Moroni, and Laura Tacheny provided detailed comments andgentle nudges throughout the process. Sincere thanks also to Claude Bertout,Federica Ciamponi, Craig Fraser, Tamara Frontczak, Daniele Galli, StefanoGattei, Enore Guadagnini, Pierre Henri, Michael Kinyon, Ted LaRosa, IlanLevine, Monika Lynker, Patrice Poinsotte, Veronica Prada Moroni, Paolo Rossi,Gabriele Torelli, Roberto Vergara Caffarelli, Lesley Walker, and Lyle Zynda foradvice and many valued discussions about the topics in this book. AntonellaGasperini, at INAF-Arcetri, Firenze, and Gabriella Benedetti, Dipartimento diFisica, Pisa, were always ready and willing to provide bibliographic help. I alsothank Oliver Darrigol and Tilman Sauer for correspondence. Kevin Dowling, aseditor for Greenwood Press, displayed the kindness and patience of a saint andthe wisdom of a sage as this process has extended past every deadline, and Ialso thank Sweety Singh, project manager at Aptara Inc., for her valuable helpduring the publishing process. I’m grateful to Edwin and Anne DeWindt for theirfriendship, counsel, and inspiration as historians and scholars.

And thank you Cody.

This book is dedicated to the memory of two great scholars and teachers withwhom I had the privilege of studying during my graduate studies at the Universityof Toronto, Fr. James Weisheipl and Prof. John Abrams. They lived their subjectand through their love of learning transported their students into a timeless world.They instilled a critical spirit through their questions, discourses, and glosses,always emphasizing how to see the world through the eyes of the past. The long

xvi Acknowledgments

road of science is a tangle of errors, commentaries, corrections, and insights, ofmisapprehensions and clarifications, and most of all a communal adventure that Ithink is well summarized in a passage from Adam Bede by George Eliot:

Nature has her language, and she is not unveracious; but we don’t know all theintricacies of her syntax just yet, and in a hasty reading we may happen to extractthe very opposite of her real meaning.

1

FORCE IN THE ANCIENT ANDCLASSICAL WORLD

But Nature flies from the infinite, for the infinite is unending or imperfect, andNature ever seeks an end.

—Aristotle, Generation of Animals

Although the civilizations of Mesopotamia and Egypt certainly knew and used el-ementary force concepts, at least as they were needed for what we would now callengineering, they appear to never have systematized this experience within a the-oretical framework comparable to their developments of geometry and astronomy.For this reason I’ll begin with the Greeks. They were the first to attempt to forma unified explanatory physics—a science of Nature—that combined quantitativescaling methods with a set of more or less mundane principles, taken as axioms,that produced a universal tradition of extraordinary persistence and elaboration.It’s precisely this “ordinariness,” you could say “common senseness,” of the rulesthat appeared so persuasive, as we’ll see, and that also made Greek physics soincredibly difficult to supersede. Indeed, it can be argued that several centuries’pedagogy has not yet led to its eradication.

For the ancient authors, the problem of motion was the problem of force. Butthere is a sort of historical determinism in this statement: for the whole subsequenthistory the problems of mechanics have been intimately connected with how todescribe time, space, and motion. Since we’ll begin with classical and Medievalarguments, it is easy to think these are quaint and almost empty disputes andconjectures from a distant past about things that are obvious to anyone. One of thepoints of this book is to show they are neither obvious nor distant.

It seems so obvious: you push, pull, or twist something and you’ve applied aforce. It reacts: it moves, it deforms, it swerves. But why does the thing you’re“forcing” react? How does it react? Under what conditions will it react? And if itdoesn’t, why? Add to this the problem of motion—what it means to say somethingis “changing” or “displacing”—and you have the catalog of the fundamental

2 Forces in Physics

questions. There was also a theological motivation and it had to do with the stateof the world as it was created. Recall the statement, for instance, in Genesis, “itwas good.” That which is “good” is perfect and that which is perfect is unchanged.If the world created by the gods were in harmony then it should be in balance,in equilibrium. The problem is the world isn’t static, we see motion and change.Things form, creatures are born and die, the seasons pass and the Sun movesacross the sky. Either all these changes are illusory or they have a cause. To makematters more difficult, the rate of change isn’t the same among all things. Changesof location are different than changes of state. Things move in space at differentspeeds, and in various directions, and age—that is, move in time—at differentrates. Living things are automotive, acting or changing because of innate abilitiesand desires. Inanimate bodies also move, but here the cause is more obscure. Itseems to require some external agent, but it isn’t obvious that the reaction of thebody to this agent’s actions will always be the same. If you think about this fora moment, you’ll see why the problem of motion—and therefore of force—wasso difficult to resolve for so long. You need to recognize that Nature isn’t alwaysanimate and, then, that it isn’t capricious or malevolent. You also have to findsome way to distinguish between “voluntary” and “obligatory” actions and thenyou can begin asking how and why these occur.

How can we describe change? It seems to require memory, that from one instantto the next we recall that something had a previous state (location, appearance,etc.) and now different. The cognitive act, recalling a previous state and thenseeing that the thing in question has moved, is central to our notion of causalityand is a hard-learned lesson in childhood. The infant learns to recognize what is anessential property of a body, what makes something the same as it was regardlessof how its location changes. This notion of the stability of things and will becomeprogressively more important. To say something changes spontaneously, withoutapparent reason or cause, is to say it acts inscrutably. This is often applied topeople, who have ideas as agents for their actions, and in a supernatural world itcan also apply to things. To see that, instead, there are systematic behaviors—“laws” of nature—was an enormous step.

Let’s start with a simple example. Imagine a stone lying on the ground. Now itjust stays there unless you physically do something to it. Telling it to jump into yourhand doesn’t work, except perhaps in animated cartoons. Its place must be activelychanged. So let’s say, you lift it. Now you notice it feels heavy. You can move yourhand in the same way without the stone and lift the air, but the difference in weightis obvious. Now you release the stone. It falls. Obviously there’s nothing unusualabout that, right? But think again. To get the stone to rise, you had to do something.What are you doing when the stone falls? Nothing, that’s the problem! If the stoneisn’t alive, how does it “know” where to go. An even greater puzzle comes fromthrowing it upward. Again you are the active agent but the stone slows and thenreverses its direction once it leaves your hand. Why does it continue moving andwhat makes it “change its mind”? To complicate the picture, at the next level wecan ask why some objects fall—the rock on the edge of a table, for instance, ifit’s disturbed—while others never do—fire, for example. Some things fall fasterthan others, so it seems how motion occurs is somehow linked to the nature ofmatter. For all ancient peoples, the gods were always available to explain the

Force in the Ancient and Classical World 3

big events, from geopolitics to natural catastrophes, but for such ordinary thingsof so little importance it seemed ultimately ridiculous to require their constantattention to each attribute of the material world. Instead, things themselves canbe imbued with motives and potentials. You can substitute here the construct ofa “world soul” and get closer to the idea: spirits are linked somehow through asense of purpose to produce the required state of the world according to the masterplan. The Platonic notion of perfect forms, and of the Demiurge, the purposefulcreator who sets the whole world out with a reason and then steps back knowinghow it will develop any why, is enough to explain anything without contradictions.If something seems extraordinary, for better or worse, it can be put down to theparts of the design that are unknowable by humans and therefore simply therewithout requiring deeper probing. This deistic worldview has no problem withany cause because it is inherent to the thing in question having been carefullyprovided by the design at the start. The remaining problem of first cause, theorigin of the Designer, is left as an exercise for the reader, although we will soonsee that this too produced problems and resolutions in the original mechanicalsystems.

Starting from a purposeful world, motion of a body is, in itself, not mysterious.For instance, if a living creature wants to move, it moves. There is some internalanimation, caused by the “mind” or “will” which, when coupled to the “machine”of the body, produces a change in the being’s state of rest or motion. If the worldwere animated, if there were a “world soul” as the agent for change in the naturalworld, then motion would always intentional and, one might say, reasonable.That’s all that’s required in a supernatural context. But it becomes strange whenwillful cause is replaced by some action of the world on the thing in questionwithout “motivation.” We call this new concept Force and the boundary betweenspontaneous and willful motion and obligatory response to the “laws of nature”marks our starting point in this journey. Force as a causative agent to motion isthe core concept of classical physics. No other study illuminates so clearly theconfrontation of experience with the physical world.

THE SETTING OF ANCIENT PHYSICS

The Greeks were the first civilization to develop the notion that Nature is com-prehensible through logic and based on only a limited number of principles.Starting with Thales, around 700 BC, they were drawn to philosophical—thatis, non-theistic—explanations of phenomena in the world around them. Beforethe fifth century BC, Anaxagoras and Anaximander were performing demonstra-tions from which they drew physical conclusions, the beginnings of empiricism,and Pythagoras had applied mathematical reasoning to the discovery that musi-cal harmonies are related to number. Among the philosophers before Plato, thePre-Socratics, we find a wide range of speculation about causes. None of theseare more than suggestions, they were frequently couched in metaphor and anal-ogy. Even Plato, in the fourth century BC, dealt more with perception, logic, andmathematical reasoning—the discovery of abstraction as a way of understandinghow to penetrate beneath appearances to the nature of things—than with physicalcauses.

4 Forces in Physics

At Athens, Plato attracted remarkable talents to the Academy. His circle in-cluded the geometers Euclid and Eudoxus. And then there was Aristotle, whowill be the focus of our discussion of ancient Western physical thought. Thereare several reasons for this. He was the first to create a systematic body of worksextending across the whole spectrum of physical experience, the first to develop acomprehensive method of analysis—the dialectic—and the most complete sourcewe have for his predecessors. But it is perhaps more important that later writersbegan with Aristotle. For the Scholastics, he was “the Philosopher,” the font oftheir methodology and the foil of their arguments. He was the source for the majorproblems of physics, among many others, and his is the largest surviving collectionof treatises on physical subjects from the Hellenistic age.

NATURE AND MOTION: THE PHYSICSAND DE CAELO OF ARISTOTLE

A characteristic feature of ancient natural philosophy that, persisted into theRenaissance, was a preoccupation with classification—taxonomy—by which athing can be described and consequently known. This approach, which worksremarkably well for zoology and botany, was one of the inspirations for Aristotle’sapproach. You first classify the thing being studied. For example, you ask “whatkind of motion are we seeing,” the idea that there are different “species” of aphysical phenomenon is at the base of the method. Then following the enumerationof the detailed properties of the subject, through a series of “compare and contrast”exercises, you arrive at its essential properties.

Two works, Physics (“On Nature”) and de Caelo (“On the Heavens”), presentAristotle’s ideas about earthly and celestial mechanics. Above all, for him motionis the problem of force, and vice versa, and thus it remained for nearly two thousandyears. Aristotle’s way of explaining motion can be viewed as a transformation ofchoice and motivation into physical necessity. His mechanical ideas were setwithin the broader context of taxonomy: to find the agents by defining the actions.Thus the basic distinctions among the causes is between those that are intrinsic tothe body and those that relate to its state. These causes, the fundamental elementsof Aristotle’s explanations, are enumerated very early in the Physics, in the secondbook:

� Material: anything that exists must be composed of something material, so the stuffout of which a body is produced contains something causative.

� Formal: the thing also has some form, some shape, some components, and these arecausative as well. Formed differently, the material will act differently.

� Efficient or effective: something or someone creates the thing, so there is a causationin this. Here we meet for the first time intention in the creative act, and at the end ofthe sequence we have

� Final: the ultimate, purposeful end toward which the thing acts. In other words, onceformed, there is a final state that the thing should be in.

As a naturalist, Aristotle took his analogies from the living world. It is, therefore,not surprising that within the four causes, the material has the potential withinitself to produce any action if it is properly “stimulated.” This latency becomesa property of a substance. How this metamorphosed into a universal, far more


general idea of “potential” we will see once we get to the eighteenth century. Butin its ancient context, this “Potency” is very different from a force, or mass. It is acause within the thing, something proper to the nature of the object that can eveninclude its form as well as its substance. While the action isn’t obviously voluntary,it almost resembles a magical property, an animate nature, that if understood canbe controlled and that stays with the individual thing. There is nothing universalabout the things we see in the world except their composition: if you take thebasic elements, that have specific properties in themselves, and combine them toform the material world, the things thus formed will retain mixtures of those basic“natures.”

In order, then, to understand how things move and change it’s necessary to alsohave some idea of what matter is and how these properties mix within something.For Aristotle, there were four fundamental elements: earth, air, fire, and water.Each possesses specific, distinct properties that imparted certain predictablebehaviors to matter. Substance is continuous, there are no fundamental indivisibleparticles, atoms, from which everything is constructed, and the essential ingredientis the mix of these elements. Time is a “magnitude divisible into magnitudes,” ashe affirms in Book 3. Any interval can be infinitely subdivided; since the presentis an instant, there must be an infinite past and an infinite future. The same holdsfor space and, by combination of space and time, all motion is continuous, so onlya finite motion can occur in a finite time. Like magnitude, or line in geometry,motion is infinitely divisible. In its division, there are two distinct types of motion,“violent” or “accidental” and “natural” that are essentially different. All matter iseither simple or compound in nature depending on the mix of the four elements,each is dominated by the appropriate natural motion of the principal components.If you imagine the analogy with living things, this is very intuitive. The taxonomy isbased on recognizing the proportional distinctions between creatures and plants.But this also applies to inanimate things. For example, you can imagine a bodythat is water and earth—that is, mud. If dropped, it will follow the path of thedominant element, earth, and seek it’s appropriate place, the center. A cloud canshoot fire, rain, and rise so it too must be a compound of fire, water, and air.

The reasoning is actually remarkably successful in distinguishing among thethings of the material world. It even seems predictive of their behaviors. Eachof the elements has its natural motion and its necessary destination of a naturalplace. These might, however, be the same. For instance, although air is transparentand light, it has weight (this was already well known for at least a century beforeAristotle, but fire doesn’t. Yet smoke is a mixed thing that comes from releasingthat element from a body. The heaviness and its natural place, down, permitted—even required—one more concept within Aristotle’s physics, the most sweepingconclusion of his whole system. Because all heavy things—that is, all thingsendowed with gravitas (weight)—fall to the ground, and because from anywhereon Earth you see the same action, since the Earth is a sphere it must be at thecenter of the world. From the principle of “natural” motions, it also follows that theEarth must be immobile because it must seek its own center; a substance cannothave within itself more than one natural motion so rotation or orbital motion areboth ruled out. Thus we have a complete world picture from the outset, a physicalbasis for the constructions of the astronomers and mathematicians, a cosmology.

6 Forces in Physics

Instead, “violent motion” is impressed on a body by an agent that causes it to actagainst its natural motion or that disrupts it from its natural state. Thus, any timeyou cause something to move contrary to its expected behavior the motion mustdiminish and ultimately end. While natural motion only terminates when the bodyreaches its appropriate place, violent motion stops of its own accord when theaction of the moving agent stops. A willful, even random or accidental cause canproduce any motion you’d want, but the agent can’t make something act againstit’s nature forever and its influence must be expended. The natural motion willeventually dominate.

TIME, PLACE, AND CHANGE

How do you separate the space in which a body moves from the body itself? ForAristotle, this was subtle. One needs to imagine something like an “external”space, indeed space itself is a very difficult thing to realize if you cannot imaginea vacuum. To have an empty space that is not merely an abstraction, somethingthat isn’t just a vacuum but genuinely nothing but a coordinate reference frame,requires being able to put a physical thing—like a material body—within anothingness. Then, to have this body move, we can ask if the space that containssomething is separate from the thing it contains? And if something changes itsstate, if it moves or varies in one or more properties, how is the change to bedescribed? Even more puzzling is how we know that something has changed.

All change becomes relative and takes place in time. One of Aristotle’s prede-cessors, Zeno of Elea, found this puzzling. Since something can be called differentby simple juxtaposition with another state, irrespective of when the two states oc-cur, we could imagine motion occurring in the same way. These are his (in)famousparadoxes of motion. Taking just one of his examples, the flight of an arrow, wecan ignore the cause of the motion and simply ask if it is possible for the arrow tobe moving. Of course, you’ll say, we have two positions and just compare wherethe arrow is relative to them. But for Zeno, instead of taking this external view,he poses the problem as one seen from within the moving frame. The famous, andin its time thorny, problem of “Achilles and the Tortoise” is posed this way. IfAchilles starts from point A and the Tortoise at B, then when the Tortoise movesfrom B to C, Achilles moves from A to B. Similarly with all displacements. Sincethe two points bounding the displacement are never coincident, the conclusionis the Tortoise is safe; Achilles can never capture it. But Zeno has cleverly left outthe time required to move through the respective intervals. Having space aloneisn’t enough, motion requires time: if we ask how fast Achilles moves relative tothe Tortoise we come to a very different (perhaps unfortunate) conclusion for thehapless reptile and agreement with our senses. Thus in the third book of Phys-ica, Aristotle makes time the central argument, along with the nature of actions,reaction, contact forces, and the infinite while in Book 4, he treats place, void,compulsory versus voluntary motions, and again the nature of time.

Space and time were whole in themselves, but for Aristotle there were “types”of change. Motion isn’t only spatial displacement. It could also be an alteration ofits current condition, a change in something’s state. If the world were, for instance,in equilibrium then nothing would vary. But we see change of many kinds so we


have to ask why this happens. You can see here the argument against Parmenidesand his school who, in the generations before Aristotle, had affirmed that allchange is illusory and purely a product of the senses. Instead, we now have adistinction of three types of change. First, as the term implies, accidental changesare those events that just happen, although not necessarily by chance. They canbe intentional acts or haphazard events; it suffices that they are not somethingproduced by the thing in motion on its own, nor by its essential properties. Thiscontrasts with essential change that depends on the nature of the matter, theparticular mix of characters. The last is self-motion. While it’s almost obvious forliving things, it’s the hardest for Aristotle to explain for “stuff” without resorting tosome animating principle. To flag a point now that we will soon face in more detail,when recovered in the Latin West, this Aristotelian idea became the link betweentheology and cosmology. The self-moved thing, the Primum mobile or first mover,is perpetual and perfect, and therefore was identified later with an animate God.

Ironically, it was just this identification that Aristotle and his followers, the Peri-patetics, sought to avoid. Although there are natural motions, there is a differencebetween matter in general and living things. The first are moved by principles—laws—while the second are moved by desires and intentions. Stuff, matter, ismerely obedient to laws and not capricious. The Prime Mover is also bound inaction by these laws and does not act in an intelligent way to control the system,it just does what comes naturally. Being composed of a distinct substance thanterrestrial things, it cannot be found in the stuff of the everyday world but it isconstrained to behave according to the laws that govern everything.

THE THING MOVING

Now, finally, Aristotle can treat the motion of the affected body and the nature ofcorporeal motion, whether the part or the whole is moved, and the impossibility ofmoving an infinite distance in a finite time. Again, these are set within a specificpicture of how the force is transmitted to the object which is always through somecontact force.

In book 7, Aristotle makes the statement that would echo down the centuriesas the axiom of motion. He states that everything that moves is moved by another.For if it does not have within itself the power of moving, it is evidently moved byanother. Although this seems circular at first, it is an essential separation betweenthings that have a natural motion or are animated and those that experience violentor accidental motions because something outside has started it and maintains it.Freely falling bodies do not have an obvious driving agent yet they move—this wasthe point about something having within itself the power of motion. For example,imagine a rock sitting on a shelf. If the shelf breaks, the impediment to the naturalmotion is also removed so the motion starts. It continues to fall until the groundgets in the way and stops the motion. The rock comes to rest because it isn’t able toovercome the resistance of the surface. Both of these depend on circumstance, sothey’re accidental in Aristotle’s view. But the rock, being heavy, contains withinitself the natural cause: it seeks the center of heaviness, the center of the world.In contrast, an animal, is born (comes into being) with its willful ability to moveas part of its nature. No external driving is required.

8 Forces in Physics

How the mover acts and what it ultimately is were questions that Aristotle couldnot avoid and in the Physics and in de Caelo, it is at the core of cosmology towardwhich the whole of the Physics has been directed.1 Thus, in the last section ofthe ultimate (eighth) book of Physics, Aristotle at last tackles the problem of firstmotions and their initiation. If all motions require something else to initiate them,there must be a first agent or “Prime Mover.” The theological and philosophicalimplications of his solution dominated physical theory for almost two thousandyears.

CELESTIAL MOTIONS

The problem of planetary motion was distinct. Unlike the stuff of everyday life, theplanets—and this includes the Sun and Moon—revolve unceasingly. The starsnever change their motions, staying fixed relative to each other and revolving fromeast to west daily. They neither fall nor slow down. The planets, on the other hand,are more complicated. They too follow the daily motion of the stars but displacerelative to them.2 Most of the time they move in the same direction as the Sun,opposite the rotation of the stars, from west to east along the same path followed bythe Sun. But occasionally, periodically, they reverse this motion and pass througha short period of retrograde during which they stop, reverse their angular motionfor a period of weeks to months, stop again, and then continue as before. Theangular motion is not constant even in the prograde part of the revolution. Forthe Moon and Sun, the variation is substantial, for the others (except Mercury, forwhich the motions are especially complicated) it is barely observable and requireda long period of observations and tables. The irregularities of the lunar and solarmotions were already well known to the earliest astronomers, the Babyloniansand Egyptians, had codified the kinematics in a set of laws from which theycould construct tables. The Babylonians had developed a system for computingthe solar and lunar wanderings among the fixed stars, treating the rates of motionas a function of time through simple algorithms that changed the angular speeddepending on the position relative to the Sun. But notably, neither they nor theEgyptians hypothesized any physical explanation for these motions. If the tablesworked well enough to cast horoscopes and determine time, that was enough. Asto their essential natures, celestial bodies remained inscrutable.

The Greeks were the first to combine these individual cases into a singleexplanatory physical system. Eudoxus, a contemporary of Plato and Euclid in thefourth century BC, is credited with the first kinematic explanation for the motionsusing geometrical methods. First, he assumed all motions are centered on the Earththat sits unmoved at the center of the sphere of the fixed stars. This sphere rotatesonce per day around a fixed axis. Then he asserted that the mobiles are carriedon nested, variously tilted geocentric spheres; the number varied depending onwhat was required to reproduce the motion of each planet. The model, and here weencounter—really for the first time—an attempt to create a general explanationof all possible motions of the celestial bodies, required alternating directions ofrotation so the combined effect could not only produce the normal west-to-eastmotion of the planets relative to the stars but also explain the daily motion ofthe sky as a whole and the occasional reversals of the planetary progressions.


These intervals of retrograde travel cannot be accounted for without compoundmotions.

Aristotle adopted this system essentially unaltered but sought to add a drivingmechanism. Not content to merely mimic the motions, he sought a cause andfound it in the contact of rigid spheres with a complicated analog of a transmissionsystem of a machine. Sets of compensating spheres, increasing in number as newphenomena were included, could reproduce the basic kinematics while remainingconsistent with the dynamics. Most important, since the “infinite” is inadmissiblein Hellenistic cosmology, there must be something maintaining the driving ata finite distance from the center. This was the primum mobile or Prime Mover.For Aristotle this was the outer sphere was the origin of the motions and hadwithin it the formal and material causes for driving the motion of those sphereswithin it.

STATICS: ARCHIMEDES

If the science of motion was the stuff of philosophy within mechanics, statics wasits applied side. The development of technology had always been practical and bytrial and error, with artisans learning the hard way how to produce, and then repro-duce, mechanical effects. Machines were inherited and modified conservatively,

Figure 1.1: A Roman mosiac from the first century AD, discovered at Pompei, depicting the deathof Archimedes during the Roman siege and conquest of Syracuse. Image copyright History ofScience Collections, University of Oklahoma Libraries.

10 Forces in Physics

angles are identical

L

L′

∆′

∆

Figure 1.2: The level. Displacement of the left downward movesmore mass at a larger distance from the fulcrum (the supportpoint) through the same angle than the right side (the verticaldistance on the left is greater than that on the right).

almost like natural selection. Oncea function was fixed, they were var-ied only hesitantly. The builderswere known by their skills, whichwere acquired by experience. Butthe Greek philosophers were aquestioning bunch and even tech-nical matters excited their curios-ity. The systematics of these rulesof thumb were like laws of nature.Mechanical contrivances were, af-ter all, operating according to me-

chanical principles and it became a challenge to find out what those were. Aristotledidn’t say much about the equilibrium of continuous media in the Physics. Thepoint of the work was to examine motion (although in book 4 of de Caelo, he dealtbriefly with floating bodies). A separate tradition began with the Mechanical Prob-lems, in which equilibrium and change were treated together. This work, whichis no longer considered Aristotle’s own but that of one of his students (proba-bly Theodorus, who followed him as head of the Lyceum) dates from the periodimmediately after Aristotle, at the end of the fourth century BC. Its schematicpresentation resembles lecture notes expanding on basic issues from the Physicsand relatively brief responses to a wide range of mechanical questions. The workwas unknown in the Latin West but it was quite popular among Islamic scholarsand consequently indirectly exerted a broad influence on doctrines that wouldlater jump the Bosporus into Western arguments.

The principal developer of these ideas in the ancient world was Archimedes,who was born in Syracuse in 287 BC and died during the Roman siege of the city in212–211 BC. His was a unique combination of talents in mathematics and appliedmechanics, what we would now think of as the prerequisites for being a successfulengineer. But I’ll call him the first “physicist” in a sense very different than theother Greek philosophers of nature. He had the peculiar ability to extract broadtheoretical principles from mundane problems and to then turn these around toproduce devices and practical applications. His particular concern was not withmotion but statics. In effect, he was the first to see rest, not just motion, as animportant subject of inquiry and the first to apply quantitative, mathematicalreasoning to bodies in equilibrium.

Aristotle had asserted that once a body is in its natural place it is at rest and,therefore, in equilibrium. Instead, for Archimedes, equilibrium isn’t somethingthat just happens. It’s when opposing forces, whatever that means, balance. So it’snot surprising that the balance, or more precisely the lever, provided the necessaryexample for Archimedes. Our main source for his thoughts, as it was for the Islamicand Medieval philosophers, is his On the Equilibrium of Planes or the Centers ofGravity of Planes. His approach was deductive, although his axioms were those ofan applied natural philosopher and geometer. His presentation is worth examiningat some length. He begins with a set of axioms the principal one of which is“Equal weights at equal distances are in equilibrium and if not equal, or if atunequal distances, the greater weight if unequal or the one at the greatest distance


if equal will displace the plane downward.” Then he presents essentially the sameassumption but with a new condition, what happens when a new weight is added tothe system so it goes out of balance and then uses more or less the same but now ifa weight is subtracted. This is typical also of the later presentations in the MiddleAges, each case is stated separately even if they amount to the same thing withchanges in the signs (for instance, a weight subtracted is a negative weight added, adifficult concept before algebra). He then assumes that “Equal and similar figurescoincide applied to each other as do their centers of gravity” and follows the sameprocedure asserting that “In figures unequal but similar the centers of gravity aresimilarly placed. Here, however, an additional construction is added to the axiom,that if points are similarly situated and connected by similar lines they will be inequilibrium. A fundamental new element is added when he proposes to quantifythe arguments by introducing the word “magnitude” to indicate the weight of abody. Finally, the location of the center of gravity must be within the figure if itssides are concave in the same direction. This last axiom, again purely geometric,provided a very important notion in his studies. With this assumption Archimedescould specify how to distinguish between “interior” and “exterior” components ofa body.

Archimedes then extends these axioms in the propositions using the style ofgeometrical reasoning, gradually building the general result by examining specialcases. For instance, the third proposition, unequal weights will balance at unequaldistances if the distances from the fulcrum are inversely proportional, is the basiclaw of the lever. Prop. 4 combines figures to determine their common center ofgravity if the figures are self similar or joined by a line to their respective centersof gravity. Prop. 5 extends this to three bodies along the line connecting thecenters of gravity. Prop. 6 and 7 restate, more precisely, prop. 3 now not usingcommensurate figures (in other words, here he uses the notion of dead weight).Prop. 8 is the same as the level for the interior of a figure, thus beginning thedefinition of equilibrium of a solid within itself. prop. 9 deals with the equilib-rium of parallelograms, Prop. 10 makes this precise to the statement that thecenter of gravity is at the center of the figure (intersection of the diagonals ofthe parallelogram). Prop. 11 deals with similar triangular planes, extended to thelocation of the center of gravity more precisely in prop. 12 and 13. These aresummarized with prop. 14, that the center of gravity of any triangle is at the in-tersection of the lines drawn from any two angles to the midpoint of the oppositesides, respectively. This was later extended in the Method. The final prop. 15, isthe culmination of the triangular figure but makes use of the parallelogram con-struction. The second book opens with the consideration of parabolic segmentsand then parallels the presentation of the first book. In effect, having establishedthe construction for simple figures, the point of the second book is to demon-strate its extension to arbitrary, curvilinear figures of which the parabola happensto be an analytically tractable case. Thus, using purely geometric constructions,Archimedes was able to generalize the treatment of balance to a broader class ofbodies.

The importance of his concept of equilibrium is more evident in his nextwork, as far as they can be sequenced chronologically, On Bodies Floating inWater. The moment of discovery is the stuff of legend, told by Vitruvius, the


first century Roman mechanician—Archimedes noticing the rise and fall of thelevel of water in a bath as he lowered or raise himself and in a fit of excitement,running naked through the streets of Syracuse yelling “eureka” (I’ve found it).Whatever the truth of the story, the observation was correct and profound andsufficient to solve the problem of how to determine the density of an object moreprecisely than could be affected with a simple beam balance. In his theoreticaltreatment, Archimedes discusses the application of a balanced state of rigid bodiesimmersed in a deformable incompressible substance, liquid water. The treatmentis a beautiful application of the lever, taking the center of mass of a figure, inthis case a paraboloid (a surrogate for a boat), and treating the limiting cases ofstability. For the first time, we are presented with constraints within which a bodycan maintain its balance. This is not, however, a stability argument. Archimedesdoesn’t discuss how a body reacts, or how it regains its equilibrium, or whether itcontinues to move away from that state, nor what happens to its motion as it does,but only under what conditions the body will reestablish its balance. In all cases,there is no discussion of the motion resulting from thrusts and drives. The newprinciple is relative weight, the free air weight of a body as measured by a balancereduced by the weight of the water displaced when it is immersed.

A material object in balance under its own weight is described in terms onlyof gravity and moment. Weights or magnitudes, which are the equivalent in theArchimedian literature, generalizes the results to extended bodies. The centerof gravity becomes not the center of figure, which may take any shape, but thecenter around which the figure is in equilibrium and that consequently producesits entire effect as a weight. While this seems simple, it was a distinct change fromthe classification of formal causes that dominated much of the rest of physics. Thebody can have its weight redistributed in any manner along an equilibrium line aslong as it doesn’t move its center of gravity. Archimedes uses this in the Methodand also in the two mechanical works on equilibrium.

This is the part of the mechanical tradition that intersects experience, especiallyregarding the construction and stability of structures. Archimedes applied it to awide range of common problems, from the balancing of beams to the equilibriumof floating bodies. In this second, he stands as the founder of naval architecture.Not only are we dealing with the ability of a body to stay afloat, that follows froman application of the equilibrium principle of the level, but its ability to stayupright dependent on shape and loading can now be addressed. Here the lever isparamount: the location of the center of gravity and the length of the moment armdetermine whether any body can remain stable.

Not only machines were now the object of physical investigation. Even con-struction was encompassed in the Archimedian approach to materials. A beam isjust a lever and an arch is a distributed load along a curved beam. The monumentsconstructed by the Egyptians, Greeks, Romans, Chinese, and Indian builders havesurvived millennia, a testament to the impressive body of empirical knowledgeaccumulated by ancient and Medieval architects. No theoretical treatises existon any of these structures. The Hellenistic compilation by Hero of Alexandria,dealing with mechanical inventions along the lines of Archimedes, and the laterRoman work of Vitruvius and Medieval Vuillard di Harencourt describe a widevariety of machines and designs but generally with no need to invoke calculations.Based on the simplest geometric forms, since the builders were unschooled in


geometry and could deal easily only with right angles, circles, and polygonal in-scribed or circumscribed figures, nonetheless yielded a rich heritage of structuresand architectural forms. This lack of theoretical basis for the building process wasno barrier to the realization of the projects. It meant a skilled class of artisansdeveloped over the centuries and that the accumulated experience formed a tra-dition rather than a formalized body of knowledge. Nonetheless, this provided abackground for many of the later discussions.3

Applied Hellenistic Mechanics: The Basic Machines

When Archimedes wrote about thrust and driving, he was not using the terms intheir modern sense (although it sometimes seems so from translations). Instead,the thrust is the weight of the body and the reaction is that of the fluid. Becausethe medium, water, is incompressible there isn’t any need to discuss the structureof matter or any effects of the actions on the bodies themselves. In this sense,he could deal geometrically with equilibria. But lacking any driver, or it seemsany interest in what happens outside of balanced forces, he was content to dealonly with those configurations that are stable. For instance, in the treatment of therighting of a floating paraboloid, he solved a basic problem of naval architecture—where to locate the center of mass of a vessel and under what conditions it wouldbe stable (depending on the load), but what happens if things go out of kilter isnot Archimedes’ problem.

That is more a problem for those we would now call engineers, the artisans andtechnicians who actually made real working things. The Hellenistic mechaniciansof the first century, especially Hero of Alexandria, and their Roman successors,especially Vitruvius, began their discussions of mechanisms in much the samefashion as Aristotle divided the causes in his discussion of motion, by distinguish-ing five from which virtually all others derived. These were the block and tackle orpully, the wedge, the screw, the winch, and most important of all, the lever. For all,the same principle applies, mechanical advantage is the efficiency with which asmaller force can overcome a larger one. The lever provided the main example ofequilibrium of extended bodies, the pulley showed how a force can be distributed.Again, all that was needed was weight (loads) and reaction (tension) without anyclear idea of forces. For instance, a cord provides the tension to lift a body and,in general, this was treated without extension. A single wheel suspended from arigid beam or ceiling allows you to redirect the force but changes nothing aboutits magnitude—you still need to exert W units of force for each W units of weight.But if a second pulley is added and has a weight attached to it, then the cordextending over the two with one end attached to a rigid support reduces by a factorof two the amount of force required to lift W . Adding more wheels to the systemfurther reduces the required force, each provides a factor of two reduction. Thiswas described by Hero in his Mechanics and became the block and tackle. For theother mechanical elements the treatments are similar.

THE BIRTH OF THEORETICAL ASTRONOMY

Hellenistic science achieved a complete picture of natural phenomena in theterrestrial realm within Aristotle’s physical system. But the motions of the Moon,Sun, planets, and the stars presented different but related difficulties to those in


the nearer parts of the world. Let’s begin with the sky. The stars move continuallyin the same direction and, apparently, with the same angular motion. But nothingelse does. This perpetual motion is completely alien to the physics we have justoutlined. Aristotle and those who followed saw at least one immediate solution tothis dilemma. If the universe is governed by the same causes everywhere, theymight differ in their effects depending on site. So it was straightforward to addanother postulate to the system. Here around us, we see things changing. Weatherprovides a perfect example. Clouds form and dissipate, it rains, there are windsfrom various directions at different times. While these must originate from thesame principles as the abstractions of single particle motion and equilibrium, it isalso obvious that the celestial bodies don’t show this same lack of predictability.There is nothing of chance in the heavens. This can be included in a generalscheme by separating the universe into two parts, those within the atmosphere andthose without.

Although the Moon changes shape, this was obviously due to its relation tothe Sun and merely an effect of reflection of light. Otherwise it behaved like theSun and planets, moving perpetually around the Earth and relative to the stars.Because the Moon eclipses the Sun and occults stars and planets, it obviouslylies closer than they do. Its distance was known from direct parallax measure-ments, at least in Earth radii, before the start of the first century and as earlyas beginning of the third century BC Archimedes and Aristarchus of Samos hadproposed lower limits for the distance to the Sun. The sphericity of the Earth andits central location were supported by astronomical observations, and even therelative size of the lunar to the solar orbit could be estimated from the durationof lunar eclipses. The fixed stars, however, were a different issue. They couldn’tbe infinitely far away; Aristotle had argued, sufficiently convincingly, that an in-finitely fast motion cannot occur in nature and since the heavens revolve in afinite time they cannot be infinite in extent. But unlike the Moon, and the Sun,no simple geometric measurement or argument could furnish a stellar distancescale. It was therefore reasonable to proceed by taking the Eudoxian spheres asthe fundamental mechanism, for planetary motion and make the outermost spherethat transports the stars and planets in their daily motion the outermost of theuniverse.

Celestial Motions

The Classical universe was divided into two parts: “down here” and “up there,”the things of everyday life and the things in the heavens. Nothing connected themdirectly, leaving aside the notion of “influences” that served as the wrappingfor astrological prognostication.4 The distinction between the two realms wasalmost tautological. The celestial was eternal, the terrestrial wasn’t. If one of theseheavenly things fell, for instance meteors, it was reclassified. Any motion thatseemed to share properties between the two regions, comets for example, werethought to be essentially something terrestrial in origin in which, somehow, theproperties were mixed. Freefall and celestial circulation were treated differentlybut both were, in Aristotle’s sense, natural since they were “proper and essential”to the bodies that display these motions.


Circular motion required a special treatment. While it was straightforward topostulate a teleology for the fall of a body, or some similar explanation for the lossof impulse in violent motion with increasing distance from the source, circularmotion and its continuation posed more complex problems. It required insteadformal and material causes to sustain the motion, the perfection of the circle (orsphere) and the nature of the substance from which the celestial realm is formed.The separation of the heavens from the Earth made it plausible that the matterwould also be distinctive and therefore able to do things you wouldn’t expect tosee happening in the world below. This separation, which is an essential featureof all pre-modern physics, was reasonable in the absence of general principles forthe initiation of motion and the transmission of action.

Freefall provides the empirical basis for cosmology and also the context forexplaining the continued circular motions of the planets and other extraterrestrialbodies: the universe is centered on the Earth, which stands immobile. This was astrue for the mechanicians as it was for the astronomers. Archimedes, for instance,took this for granted and used it to explain the equilibrium of layers of fluid.

You should distinguish the dynamical and kinematic explanations. For de-scribing the structure of the world, cosmology needs a consistent principle. Themotion is the only thing that matters: you don’t need to ask dynamical questionsbecause all motions are identical, simply circular. Thus the kinematics reduce tojust geometry, albeit with the addition of time as a reference. The relative motionsare important for astronomical observation and the chosen reference is to the Sun.

A few months’ observation, if properly timed around a planetary oppositionwhen the planet is in the part of the sky opposite to the Sun (meaning we restrictattention to Mars, Jupiter, and Saturn in order of their periods) is enough to seethat these bodies exhibit deviations from a steady progression along the ecliptic.These epochs of retrograde motion, again being purely kinematic, can be reducedto geometric analysis as Apollonius (third century BC) was the first to clearlydemonstrate. The kinematical proof makes no use of the concept of acceleration.All apparently irregularities must be due to compounded circular periodic motions.How these add is a separate matter from their origin. The motions are continuous,it’s only our particular vantage point that makes them seem to be accelerating.With this axiom, and the cosmological assumption of terrestrial immobility, it waspossible to construct a model of the universe and a single surviving work by theAlexandrian geometer Claudius Ptolemy (flourished mid-second century) showshow it was accomplished: the Mathematical Composition (in Greek, Syntaxis).Written in the middle of the second century, it became the most important sourcefor theoretical astronomy among the Islamic mathematicians, for whom it was the“The Great Book,” by which name it became known in the Latin West when itwas translated in the twelfth century by John of Seville: Almagest. But it isn’t withthe mathematical constructions that we’ll be concerned. This is also a work alsoprovides a record of how Aristotelian physics merged with cosmology to create aworld system for the cosmos.

Ptolemy, because he was writing a work on the motions of the planets, neededto discuss the construction of the heavens. His physical premises are stated inthe introduction to Almagest, asserting that immobility of the Earth underlies allother principles of celestial mechanics. If the planets are to continue in movement,


they require constant driving: they move because they are moved. The mechanismis contact, the agent is spherical, and the driver is external and perpetual. ForAristotle, and for all subsequent commentators the ultimate cause for the motionwas a property, or principle, of the Prime Mover. In this system, to find a causefor the motions requires ending what would otherwise be an infinite regress. Thefirst cause must be something that has this action as part of itself. The ancientastronomers and geometers, particularly Apollonius, Hipparcos, and Ptolemy, allfaced a problem. The mean motions are regular. But you see deviations thatare systematic and periodic in both space and time. These behaviors could berepresented by successively approximating the motions, by adding motion-specificmechanisms to the orbit in a way that produced ever more refined periodic timeseries for the kinematics.

Let’s take a moment to examine how this construction worked. By assertion, anymotion must be either regular or compounded from regular motions. This holdsfor the periods as for the directions and rates of motion. Since a body can’t act“contrarily” to its nature or capriciously, the reversal of directions and changes inthe rates of motion of the planets must be produced by a combination of regular,sometimes counteracting, motions. The first motion, the mean anomaly, is the timefor an orbit relative to the fixed stars. This was the principal circle or deferent. Ifthe motion varied in a regular way, speeding up and slowing down periodicallyaround the orbit, it sufficed to displace the observer from the center of the circle,the eccentric. If further variation was observed, other points of relative motioncould be added, the first was the equant, on the opposite side of the center fromthe eccentric. Since the Sun moves only in one direction with variable speedthroughout the year this worked quite well: the observed angular velocity variesbut the actual angular motion remains constant. But there are problems with thispicture: the planetary motions are not identical. Although the Sun and Moon moveonly from west to east in ecliptic longitude, albeit with variable speed, they neverreverse as do three of the planets: Mars, Jupiter, and Saturn. Also, relative to theSun, Mercury and Venus also reverse direction. That is, they stay locked to thesolar motion, never deviating by more than some fixed angular distance fromthe Sun despite the variation in the solar angular speed. This is the basis ofepicyclic motion. The epicycle is a circle of fixed radius and inclination thatcarries the body around a separate, moving point on the deferent.

The stationary points and reversals in direction of motion of a planetary trajec-tory is due to the relative motions adding in such a way that the body appears tobe at rest, a proportion depending only on the ratio of the relative angular motionson an epicycle and that on the deferent and not on any real change. This wasdemonstrated by Apollonius of Perga in the third century BC using a purely geo-metric construction. But the success within Ptolemy’s model comes with a price.This kinematical construction is inconsistent with the dynamical principles usedto justify its necessity: the epicycle requires motion about at least one separatelymoving point that is not simply connected to the driving spheres and, conse-quently, to the Prime Mover. This violation of cosmological construction led to aseparation of physical and mathematical astronomy very early in its developmentand was responsible for many paradoxes, some of which were later resolved by theCopernican simplification. For Ptolemy, as for other astronomers, the axioms were


physical but these could be violated by constructions if necessary. The debate hasbeen heated about the meaning of model and hypothesis in ancient astronomy butI think this much is clear. If the planets do or don’t move along epicycles wasnot really a question that concerned the ancients. On the one hand, all treatisesincluding and following the Almagest begin with a section outlining cosmology.This is a philosophical prelude. It sets the stage for the problem to be treated, butbeing mathematics—which treats ideal things—the methods used cannot explainthe motions. For that, only metaphysics (and later theology) will do. Ptolemy saysas much in the Almagest and later, in the Tetrabiblos, repeats the assertion that themodels of the heavens are aimed at describing and predicting the motions withinsome set of assumptions or rules:

But it is proper to try and fit as far as possible the simpler hypotheses to themovements of the heavens; and if this does not succeed, then any hypothesespossible. Once all the appearances are saved by the consequences of the hypotheses,why should it seem strange that such complications can come about in the movementsof the heavenly things? (Almagest, Book 13, part 2)

If these contradict, the physical picture tells us that the description is incompleteand we have yet to find the most efficient and consistent model. We need not giveup on building a picture because it is just the shadow of the “real,” what we cangrasp as a calculation is how something appears and not what it “is.” In this I thinkthe requirement of the models to preserve the appearances shouldn’t be seen withmodern hindsight. Our notion of a model has evolved through the last 500 years,separated in some ways irreversibly from the ancients. What the Renaissance leftwas a reinterpretation of Platonism and a new concept of the use of mathematicalreasoning in physical problems that in many ways doesn’t relate to those of theancients.

In particular, noting that the first mover must have a separate status in thissystem (as Aristotle made clear in Book 8 of Physics and Book 4 of de Caelo),the cause of the motion is always present. The problem was that the systemwas that each motion required a mechanism, and these were added accordingto a prescription that requires replicating the phenomena rather than remainingconcordant with the Physica. The lunar orbit is a perfect case in point. Thecombination of eccentricity—the variation in the angular speed of the Moonwithout having retrograde motions—and the systematic shift of the phase of themaximum speed forced the addition not only of an eccentric location for thecenter of motion but a second, moving point, the equant, around which the speedwas constant. Neither was associated with the center of the spheres that werepresumably driving this motion. The same was used for Mercury. Even moreseriously discrepant was the intersection of the epicycles of Mars and Venus.There was no scale in this system so the angular motions, and distances from theSun, require proper relative proportions of the deferent and the epicycle. Angularsize variations, which should have been obviously wrong to even casual observersfor the Moon, were simply ignored. So were the relative angular sizes of the Sunand Moon during eclipses. The restriction that a celestial mechanics be physicallyas well as phenomenologically self-consistent wasn’t part of the thinking.


THE STRUCTURE OF THINGS: ATOMISM AND MOTION

There was another view of the world besides the Aristotelian continuum, atomism,that provided alternative, radically different explanations of natural phenomena.The problems of the structure of matter and the origin and maintenance of motioncame together for the atomists. This line of thought, descending from Democritus(end of fifth to beginning of fourth centuries BC) and Epicurus (around 342 BC to270 BC, a late contemporary of Aristotle), proposed that material and void coexistand are everything. Matter is atomic, transient, and mutable. It’s formed fromthe temporary cohesion of atoms, which are complicated in their interactions andvarious in composition, and when this fails to be stable, the material thing decaysand returns to the dispersed medium. There is nothing other than atoms so theuniverse isn’t filled by any resisting medium.

Beginning with Democritus, but having even earlier origins, the idea that matteris based on an almost unimaginable number of different and specific, indivisibleelementary units: atoms. Our only complete exposition of this view of matter, and ofthe forces that maintains it, comes from the lyrical composition de Rerum Naturum(“On the Nature of Things”) by the Roman poet Titus Lucretius Carus, a followerof Epicurus, in the first century BC. Although we know next to nothing preciseabout him, his poem remains a rich source for the ideas that, when re-introducedin the Latin West, provoked a debate on the foundations of the world that echoes toour time. How atomism contrasted with the prevailing continuum picture is mostdramatically exhibited in the nature of space and forces. For Aristotle and hisschool, the world is filled with matter, the vacuum is impossible, and motion takesplace only because a force overcomes its resistance. For the atomists, in contrast,the universe is empty except for the myriad interacting individuals that, in theirunceasing motion, randomly unite and separate. Motion requires no explanation initself, matter is, instead, its principal result. All interactions depend on the natureof the atoms themselves, whose properties are specific to the qualities they bringto the things formed. Those of the “soul” are different than those of, say, a tree.How they interact is left unspecified, although it seems the force is thought of asimpulsive during collisions. It is enough to say the atoms are eternal and the thingsof the world are transient. That which we see is produced by chance. It’s temptingto see in this ancient idea as something similar to our worldview. We’re now soaccustomed to a discrete world at the microscopic level that the atomist doctrinethat you may feel more at home with it than that of Aristotle and the filled world.But as we will see much later, this picture is analogous to the modern conceptionof the microworld only in the broadest sense. There is no concept here of what aforce is, the motion has an origin that begins “somehow” and there is nothing hereto explain how the individual atoms differentiate to correspond to their materialproperties.

The Islamic Preservation

Hellenistic science passed into the Islamic world rather than to the West duringthe long period of political and social instability that characterized the dissolutionof the Roman empire at the end of the fifth century and the establishment of stablekingdoms and communes after the Ottonians in Germany and the Carolingians


before the start of the millennium. Although a few works survived and were pre-served in a few copies, it is clear that the intellectual fervent of the Islamic courtsand centers of learning were debating and extending the ideas and techniquesof this inheritance, especially in Spain. The philosophers were interested in cal-culational and practical problems, much less in theoretical mechanics, and theiradvances in formal analysis and mathematical technique was enormous, but that’sa separate development from our main line. In dynamics, they generally followedthe Aristotelian precepts. Islamic mechanics used, and made significant advanceson, Archimedian results and methods. In naval design, in particular, but also indiscussions of machines and balances the principles of hydrostatic equilibrium,specific gravity, center of gravity, and the lever as a paradigmatic physical device,all found their expression in the texts. In astronomy, they elaborated the Ptolemaicsystem. Thabit ibn Quura (836–901) modified the treatment of the motion of thesphere of the fixed stars (the eighth sphere) contained in the Almagest by adding ancyclic motion to explain what was thought to be a variable rate for the precessionof the equinoxes. Ibn Sina, known in the Latin West as Avicenna, wrote aboutatomism and forces. For ballistic motion, Ibn Rashud, later known as Averroes,invoked the action of the air as a means for propelling bodies in their flight, akind of hydraulic forcing that maintained motion after a violent action. They dis-cussed many of the cosmological problems in theological and philosophical termsbut made no essential changes to the explanations. A prime example is the ex-tended discussion of astronomy in Maimonedes’ work The Guide for the Perplexed,one of the clearest available expositions in the Islamic tradition of Aristotle’sprinciples.

NOTES

1. This question of how the motion is transmitted is the most contested issue in thelong literature on the history of Aristotelian physics. Nothing in any of Aristotle’s works,however, details how the action is imparted to the body.

2. The Sun and Moon are distinct since they are the only resolved objects that executethe same periodic motion as the planets.

3. Our discussion has focused on the Hellenistic discoveries. But these were not unique.A striking parallel comes from the third century in China, as reported in the San Kuo Chih:

The son of Tshao Tshao, Chhung, was in his youth, clever and observant. Whenonly five or six years old, his understanding was that of a grown man. Once SunChhuan had an elephant, and Tshao Tshao wanted to know its weight. He asked allhis courtiers and officials, but no one could work the thing out. Chhung, however,said, “Put the elephant on a large boat and mark the water level; then the weigh anumber of heavy things and put them in the boat in their turn (until it sinks to thesame level)—compare the two and you will have it.” Tshao Tshao was very pleasedand ordered this to be done forthwith.

There were two separate weighing operations for alloys—one in air and the other in water—that were used as a test for purity as early as the Han dynasty (from the second century BC tothe third century). Also during the Han, brine testing was performed using floating bodies,for instance eggs and lotus seeds. The latter are particularly interesting because a muchlater description dating from the twelfth century records that the shape of the seed was also


used to determine the relative salinity of the water. If the seed floated upright, the brine wasa strong solution, while for horizontal equilibrium the brine was weaker. These and otherexamples discussed in the compendium on Chinese science and engineering by JosephNeedham (1962) show that the Chinese experience with hydrostatics was considerablealthough without the theoretical basis that characterized the Archimedian methodology.

4. I can’t pass this by completely. The astrological tradition flourished throughoutthe ancient world and frequently was invoked to justify astronomical investigations, forinstance that a better understanding of planetary motions would improve the accuracy ofpredictions, and the practical product of mathematical astronomy was a rich compilationover several thousand years of almanacs for such purposes. There was no contradiction inthis activity, and no separation, precisely because there was no general physical law thatprevented such correlations.

2

MEDIEVAL IDEAS OF FORCE

Each of the four elements occupies a certain position of its own assigned to it bynature: it is not found in another place, so long as no other but its own natural forceacts upon it; it is a dead body; it has no life, no perception, no spontaneous motion,and remains at rest in its natural place. When moved from its place by some externalforce, it returns towards its natural place as soon as that force ceases to operate.For the elements have the property of moving back to their place in a straight line,but they have no properties which would cause them to remain where they are, or tomove otherwise than in a straight line.—Book 72, Guide for the Perplexed, by Moses Maimonides, Friedlander tr. [1904]

For the Greeks and Romans, physics has meant the analysis and explanation ofnatural phenomena. It sometimes found supernatural applications, for instancealchemy and astrology where it seemed capable of providing precise predictions,whatever their actual success. But more often it just satisfied the philosopher’scuriosity and explained links between phenomena. With the introduction of theseideas into Islamic culture and then their transmission to the Latin West during theMiddle Ages, this changed and so did the questions. In Europe, in particular, theaudience that received Aristotle’s works and the later commentaries was concernedwith very different questions than those that had motivated their author, in fact, theywere diametrically opposite. Instead of seeing principles as things in themselves,the clerics who first read and elaborated on the works of Greek science hadideological goals in mind, seeking and finding in them a justification of a religiousdoctrine. For many centuries that followed the turn of the millennium, to discussphysics—especially mechanics and astronomy—meant discussing God.

I need to emphasize this point. Some of the problems posed in the medieval textsseem almost naive. They often mix kinematics, dynamics, and metaphysics. Forexample, whether the Eucharist can be ever present and yet consumed, whether


God can be in all places at once, whether action is direct or at a distance, howthings change and yet remain constant, were all theological as well as physicalquestions. The transformation of these questions first into grammar and logic andthen into geometry took hundreds of years of debates and small modifications.The appearance of a new school of thought in the fourteenth century, one thatbegan to create a mathematicized picture of motion and force, should be seen asan outgrowth of these more speculative questions.

THE SETTING OF PHYSICAL SCIENCE INTHE MIDDLE AGES

In the twelfth century the Latin world discovered, or rather recovered, the in-tellectual production of the Hellenistic scientists and the result was explosive.For centuries, only fragments had been known of these ideas, preserved in thoseLatin commentaries that had managed to survive almost randomly from the endof the Roman empire. Few of these were at a level above the most elementary,many were corrupted redactions, and none were sufficiently detailed to permitany extension. The period we now call the Dark Ages was hardly dark, but itwas largely illiterate and there were much more serious concerns of daily life tocontend with than would permit the leisure necessary to consider the abstrac-tions of mathematics and physical science. It’s important to consider this whenconfronting the later developments. Ours is a world so interconnected by writtenand visual communication it is likely very difficult to imagine how isolated theintellectual production was during the period before the introduction of movabletype in the West. Even in the Islamic world, more privileged in its possession ofmany of the Greek products that had been translated into Arabic early enoughto promote a flourishing of commentators and innovators, individual works werecirculated in manuscript, single copies made from single texts. Their diffusionwas slow and limited to certain environments that had access to the productionand discussion. A single work could take months to produce, depending on thecontents. The few great ancient libraries, especially that in Alexandria, had beendestroyed or dispersed, and those into whose hands the surviving texts had fallenoften could do nothing with them. Some, perhaps the best example of which isthe Method of Archimedes, had been reused as writing material or even bindingfor other folios.1 In the mid-thirteenth century, William of Moerbeke translateda huge body of Hellenistic works into Latin, especially those of Plato, Aristotle,Archimedes, and Euclid, working directly from the original Greek although not theoriginal manuscripts themselves. This made works previously unavailable acces-sible to scholars. Other scholars used Arabic or Hebrew translations of the Greekworks, still others translated original Islamic works into Latin. Wider circulationof these texts took some time, but within a century they had found their way intocollections and libraries throughout Europe.2

Another feature of the medieval texts is notable. They are frequently com-mentaries, not extensions. A scientific paper is now placed in its developmentalcontext by a brief introduction, often just a series of references to indicate thedevelopment of the idea, with the main part of the paper—or book—being new de-velopments. Novelty is now the distinguishing feature, not brilliance of exposition

Medieval Ideas of Force 23

of ideas already known. The concept of received wisdom was very important inthe Middle Ages and the texts were often approached as relics to be treated withrespect rather than criticism and correction. Every manuscript was not merely anartifact, it was almost sacred. The time required to produce a single work couldbe very long and it is clear that many of the copiers, and compilers, were unawareof the subtleties of the material with which they were dealing. The climate was notone prone to rapid innovation and the long, slow, redundant production before therise of the Scholastics and Commentators and the beginnings of the universitiesin Europe was in sharp contrast to the active Islamic production that was buildingon the Greek texts. This is especially important following the two condemnationsof teaching of Averroist and Avicennan doctrines against either Aristotle orscripture, both issued in Paris in the thirteenth century. The first, in 1210 AD,forbade the teaching of certain doctrines as true. The second, in 1277 AD, wasissued at an ecclesiastical level low enough to be chilling on the Parisian scholarsbut not damning all academics and not extending throughout Europe, Like itsdoctrinal predecessor, it only prohibited taking a list of propositions as “true,”not discussing and disputing them (as long as the outcome of the debate waspredetermined). That’s why I mentioned the structure of the commentary. Thisinsured the propositions and many, if not all, of the arguments were presentedto facilitate their refutation. Thus we know considerably more about the textsbecause the edict was simply a ban, not demanding the destruction of the works(condemned texts were burned, a perfect way to prevent the spread of the precisetext of a heretical work in an era when texts may be represented by only a singlemanuscript).

One last point before we proceed. A reason for the centuries-long silence wasnot ignorance but disinterest allied with an attempt to rigidly enforce centralizedthought and a deeply conservative hierarchy. With the rise of the universities inEurope in the thirteenth century things changed dramatically. These were no longerisolated clerical centers of textual production and discourse. Instead they wereinternational centers, collecting students and teachers from all over Europe. Youcan see this same polyglot environment now in any of their descendents all overthe world but in the two centuries from 1200 to 1400 they were still novel places,loosely organized among themselves and often competitive for students and faculty.In the same way as the cathedrals, they were seen as prizes of the communitiesin which they were sited, although not without considerable “town and gown”tensions. Their history is beyond the scope of this book but their place in the growthof science in general, and physics in particular, cannot be overestimated. Thefaculty lectures were not merely delivered but also written, copied, and diffused.The scribal houses, which in their time resembled the clusters of photocopyshops that now surround every college and university, served as depositories forthe intellectual production of the faculty and students. They received licensesfrom the university to “publish” the lectures deposited there and the universitiesthemselves preserved the notes of the courses in their libraries, which were moreaccessible than those of the monasteries. Writing materials also changed, passingfrom parchment to paper, and the decreasing costs or reproduction and growthof the literate population eager for new texts and receptive to new ideas spurredchange amid a new intellectual climate.


Of course, science was not the principal subject of study, but it was an essentialelement of the curriculum. Mathematics, specifically geometry, was seen as funda-mental to train the mind in precise reasoning along with logic. Philosophy was notonly speculative or theological but also concerned with nature. Astronomy (andastrology), medicine, anatomy and botany (and alchemy), and physics all formedsubjects of study and debate. For this reason we find a significant shift in thecenters of activity and the affiliation of scientific authors with schools. Beginningwith Salerno, Bologna, Paris, Oxford, and Cambridge, between 1200 and 1500many of these centers of learning were founded and flourished, surviving warsand plagues, economic collapses and civil unrest, changing the values, discourse,ideas, and structure of Western society. Not least important was the effect on thesciences, to which we’ll now return.

MEDIEVAL STATICS

The problem of balance was as puzzling to the Medievals as it had been tothe Greeks, especially the problem of bodies balanced in different angles or onbent levers. This had been treated by Islamic philosophers but it was at theheart of the treatises by Jordanus de Nemore (flourished around 1220 AD) whoseworks were the basis of the medieval discussion of weight and equilibrium. Theseworks introduced a new physical element, a nascent concept of work. It’s easyto imagine this as a precursor of the eighteenth century notion we will discusssoon, but although the form of the argument is similar the physics is very different.Jordanus argued this way. Beginning with the lever as a constrained motion, sincethe attached weights must remain attached and the arm remains rigid, to be inequilibrium for two different weights W1 and W2 the distances at which they areplaced from the pivot, L1 and L2, must have the inverse ratio to the weights.That is, L1/L2 = W2/W1. If displaced from the horizontal, the vertical distance(the length of the arc) will vary directly as their distances. But because these areinversely proportional to the weights, even if the lever is inclined the bodies willremain stationary. Another way of saying this is that the weights must have theproportional distances from the pivot that produces equal work when displacedfrom the horizontal in either sense.

Another very new concept, positional gravity, allowed Jordanus to solve theproblem of two coupled weights on an inclined plane, one of which is suspendedvertically and the other placed along the plane. He distinguished between this andthe natural gravity that the body has along the vertical that he showed is always thelarger value. The proof of equilibrium is very revealing. He first showed that thepositional gravity is independent of location along a constantly inclined plane. Thishe shows using a new, and dramatic, idea that comes very close to the later conceptof work. But remember, we are dealing here with statics. The displacements are nottaking place in time, although the word “velocity” is sometimes used. Instead, thisis the first instance of the method of virtual displacement. If the distance to whicha body must be raised, which is directly proportional to its distance from the pivotin balance, is proportionally greater than the ratio of the respective distances, theweight cannot be lifted.


For a moment, let’s anticipate a development that at the time of Jordanuslay several hundred years in the future. Although sometimes thought to be adynamical proof of the lever, Jordanus was using a very special setup to describethe equilibrium of weights. They’re linked, either by a cord or a lever, and thusconstrained to move in a very specific way. This is not the same as motion on aninclined plane, although it shares many of the same features. Positional gravity isindeed a component of the force, although it was always represented as a virtualvertical displacement, but it is not a force in the sense that Galileo would laterunderstand. As I noted when discussing Archimedes, how a body displaced fromits equilibrium returns or diverges from that state is not the issue. It’s enoughto say that the system is unbalanced. It is for this reason, and this alone, thatJordanus was able to resolve so many problems using only the lever as a paradigm.Constrained motion with inextensible connections requires equal and oppositedisplacements of the components. If one rests on an inclined plane, it is liftedthrough the same magnitude of distance but by the projected vertical distance, withrespect to another body displaced along some other inclined plane. Because thepositional gravity is independent of location along the plane, the initial conditioncan be to place the two weights at anywhere as long as the length of the cordremains constant.

The idea of work used here is pretty vague but it serves the purpose of quanti-fying experience: the relative effort required to raise a weight is greater for a largerdistance, or for a heavier weight raised though any fixed distance. The word relativehere is very important. In the Archimedian hydrostatic balance, it isn’t the abso-lute weight (in air) that is determined but the relative change for two materials ofthe same free weight but different density—that is, different specific gravity—thatrecognizes that the fundamental feature of the balance is that all measurementsare made relative to a standard. Since all measurements are proportional, with achosen standard (as would have been done in a market using marked weights) theproblem remains without any need to use specific units. For instance, immersionof two bodies of equal weight—in air—produces different displacements in waterand the relative densities are therefore simple to determine. In effect, in all thesecases, the lever provides the tool for understanding the general problem. Even apulley can be reduced to a treatment of mechanical advantage, the relative leverarms being the different radii.

Understanding the difference between this static conception of equilibrium andGalileo’s later dynamical approach can be illustrated by an experience you’ve hadin childhood. If you recall sitting on a swing and pumping with your legs to getstarted, the initial oscillation you experienced is what Jordanus was thinking ofas positional gravity. You’re changing the distance of a weight, your legs, alongthe axis of the pendulum, the swing has what we would now say is a variablemoment of inertia, and you’re shifting your center of gravity. This is even moreobvious when you lean into the direction of motion. But what happens on a swing issomething never discussed in the medieval works. If you hit the right frequency theamplitude of the swing continues to increase; you reach a resonance, but this wasn’trealized until nearly 500 years later. The problem of balance is not connected withdynamics even when discussed using times. For constrained motion, as long as


the time interval is fixed, displacement and velocity are the same thing and it wasonly a convenience to use the latter.

The addition of components of force would become central to Galileo’s argumentfor the action of gravity. Using this notion he also was able to discuss the effectof a balanced pair of weights with one oscillating as a pendulum without dealingwith the cause of the motion or its properties. It was enough that the change of theposition of the swinging body relative to the center of mass changed its effect on thebalance, that the angle—the position—changes the weight as felt in the coupledsystem. Thus Jordanus and his commentators could solve the perplexing problemof a bent lever by showing that only the horizontal arm, the static moment matterswhen considering the balance of the weights. The proof given in the treatises isalso easily reproduced experimentally and Jordanus even comments on the pitfallsof the demonstration.

The Scholastic treatises are notable for this new approach to how to specifywhen two bodies are in equilibrium. But it would be a very serious mistake tothink that Jordanus and his contemporaries had a more sophisticated concept offorce than weight. They were discussing, for instance, displacement, velocity, andforce equivocally. This works when you are dealing with a constrained system,such as a machine, but not for free motion. Notably, Jordanus didn’t discuss that.It wasn’t until Galileo that the two come together. But by introducing componentsof weight and a static notion of virtual work, the thirteenth and fourteenth centurytreatises on statics, printed by Peter Apian in the 1530s, provided a fertile sourcefor the development of mechanics in the subsequent centuries.

In contrast to the surviving works of Roman authors, who generally dealt withstatics and what we would now call “applied” physics, the medieval analysisof motion seems much more quantitative. It still, however, maintained the samedistinctions between kinematics and dynamics we found in the earlier treatments.The concept of a force is missing, although the word is used along with resistance.It has more the nature of cause, that is something that produces and maintains themotion, and comes in a variety of types in the same way the motion does. Force isthe continued action of a mover, either by contact or somehow, and it produces amotion precisely as long as it is applied.

SCHOLASTIC DYNAMICS: IMPETUS ANDACCELERATED MOTION

Motion was a central issue during the thirteenth and fourteenth centuries, mainlyamong two schools: at Merton College, Oxford in England, and Paris, in France.These centers of intellectual activity and international contact insured a wideaudience, both the actual attendants in the lectures and public disputes heldin the universities, and through the circulation of manuscripts emanating fromthe scriptoria connected with the universities. Further, becoming more readilyavailable during the early era of printed book production, especially because ofthe role of the university presses in the dissemination of texts, these works andtheir ideas were even more widely distributed through Europe at the end of thefifteenth century.3


Acceleration had a special fascination because it involves an alteration in therate of change in the state of the body. In a sense, all motion is, from the scholasticviewpoint, forced but not all is accelerated. The particular problem was, again,ballistics. An upwardly moving body doesn’t continue this motion. After some time(distance), it reverses and falls, continuing to move in this opposite direction with-out needing any assistance. We find a solution in the works of Jean Buridan. Bornaround 1300 AD, he lectured in Paris during 1340s and served for a time as the rec-tor of the university; his death was sometime before 1360. He was one of the mostinfluential scholastics and the founder of the Paris school of mechanics. Let’s exam-ine, for instance, some examples from Buridan’s Questions on the Eight Books of thePhysics of Aristotle, in this case Book 8, question 12. Here Buridan argues againstthe idea that motion has its source in the medium, the antiperistasus principle.

2. . . . The third experience is this: a ship drawn swiftly in the river even againstthe flow of the river, after the drawing has ceased, cannot be stopped quickly butcontinues to move for a long time. And yet a sailor on the deck does not feel any airpushing him from behind. He feels only the air in front resisting. Again, supposethat this same ship were loaded with grain or wood and a man were situated to therear of the cargo. Then if the air were of such an impetus it could push the ship alongmore strongly, the man would be pressed more violently between the cargo and theair following it. Experience shows this is false. Or at least if the ship were loadedwith grain or straw, the air following and pushing would fold the stalks that were inthe rear. This is all false.

5. For if anyone seeks why I project a stone farther than a feather, and iron or leadfitted to my hand farther than just as much wood, I answer that the cause of this isthat the reception of all forms and natural dispositions is in the matter and by reasonof the matter. Hence, by the additional amount of matter, the body can receive moreof that impetus and more intensity. Now in a dense and heavy body, all other thingsbeing equal, there is more of prime matter than in a rare and light one. Hence adense and heavy body receives more of that impetus and more intensity, just asiron can receive more calidity than wood or water of the same quantity. Moreover,a feather receives such an impetus so weakly that it [the impetus] is immediatelydestroyed by the resisting air. And so also if a light wood or heavy iron of the samevolume and shape are moved equally fast by the projector, the iron will be movedfarther because there is impressed in it a more intense impetus which is not soquickly corrupted as the lesser impetus would be corrupted. This is also the reasonwhy it is more difficult to bring to rest a large smith’s mill which is moving swiftlythan a small one.

6. From this theory also appears the cause of why the natural motion of a heavybody downward is continually accelerated. For from the beginning only the gravitywas moving it. Therefore it moved more slowly, but in moving it impressed in theheavy body an impetus. This now acting together with its gravity moves it. Thereforethe motion becomes faster; and by the amount it is faster, so the impetus is moreintense. Therefore the movement evidently becomes continually faster (translationis from Clagett 1959).

In the next three questions, Buridan outlines the nature of the impetus. First,it is the motion and moves the projectile, but it is caused by the projector because


a thing cannot produce itself. Second, that impetus is not simply additive. It isnot the local motion alone but a quality of the motion. And finally, “the impetusis a thing of permanent nature that is distinct from the local motion in which theprojectile is moved.” And “it is also probable that just as that quality is a qualitynaturally present and predisposed for moving a body in which it is impressed justas it is said that a quality impressed in iron by a magnet moves the iron to themagnet. And it is also probable that just as that quality (impetus) is impressedin the moving body along with the motion by the motor; so with the motion it isremitted, corrupted, or impeded by resistance or contrary inclination.” In the lastpart he uses the example of a stretched cord that is struck. He notes that it doesn’treturn to its initial position “but on account of the impetus it crosses beyond thenormal straight position in the contrary direction and then again returns. It doesthis many times.”

I should make a remark here about the methods employed by the Oxfordand Parisian scholars when discussing physical problems. They proposed anddiscussed several distinct force laws. But none of these were ever comparedwith experience although at the same time there was increasing attention beingpaid to the role of empiricism—that is, experience—in deciding philosophicalquestions. Even when the propositions are apparently quantitative, it is mis-leading. The numbers inserted in the examples are simply chosen out of thinair, it’s actually doubtful that many, or even any, actual experiments were per-formed by those who wrote the texts; these were arithmetic exercises to illus-trate the preconceived laws rather than measurements. For example, William ofOckham (c.1280–c.1349), who is far more famous for his logical treatises thanthose dealing with physics, nonetheless discussed aspects of the Aristoteliancosmology and mechanics, especially questions about space and time. RogerBacon (1219–c.1292) and Robert Grosseteste (c.1163–1253) also advocated ex-periment, mainly for optical phenomena, but the implications for all three was thatsome laws of nature could be discovered through experiment and not merely de-monstrated.

Merton College, Oxford was the center of dynamical theory in England. Aremarkable group of scholars collected there in the fourteenth century, the mostimportant of whom were Thomas Bradwardine (c.1390–1399), William of Ockham,Duns Scotus (c.1266–1308), Richard Swinsehead (active c.1340–1355), WilliamsHeytesbury (active c.1335), and John Dumbleton (died c.1349). We’ll focus onone aspect of their natural philosophy, the attempt to render the laws of motion in amathematical form. In this effort they treated not only those explicitly from Aristotlebut other attempts to connect motion with force from the previous centuries.Their school, collectively known as the “calculators” because of their quantitativeapproach, was to be the outstanding influence in the subsequent developmentof mechanics. Bradwardine was the founder of the research program and theauthor of its principal text, although he left Merton College in 1330s. His chiefwork, the Tractatus de Proportionum, was written in 1328. Bradwardine’s methodsextend beyond the simple kinematics of the Mertonians. He suggests severallaws of motion, none of which are precisely justified but each of which has specialcharacteristics. The first is the straightforward interpretation of Aristotle’s doctrineof resistance, the velocity is proportional to the ratio of the impressed force to the


resistance of the medium (whatever that might be). The second is a compound law,that the effect of an impressed force is magnified by the body, always in ratio tothe resistance, expressed algebraically as V ∼ (F/R)n.

We take as the fundamental dynamical principle that there must be somethingcontinuously acting to maintain a state of motion. No motion is natural, in the senseof the spheres, if it is in terrestrial stuff. So as a first assumption, the velocity is a“measure” of the force. But does that mean it is equal to the power of the mover oronly an effect? This isn’t just a semantic question: it is the core of the confusionamong the commentators in the thirteenth and fourteenth century. There were twoprinciples: the force must overcome a resistance to produce motion, and it mustcontinually overcome that resistance to maintain the motion. As Aristotle had it,there must be an excess of the motive power to the resistance. Two very differentrepresentations were proposed to quantify this notion of excess. One, due to JohnPhiloponus in the sixth century, states that the speed (motion) is determined bythe excess of the force taken as a difference to the resistance or V ∼ F − R.This form was adopted by the Islamic scholar Avempace in the twelfth centuryand was best known in the Latin west through the commentaries on the Physicsby Thomas Aquinas, Albertus Magnus, William of Ockham, and Dun Scotus.They also presented the alternative construction, due to Averroes, that translatedexcess as a ratio, that is V ∼ F/R. While for Avempace, the resistance can bediminished without limit such that the force is needed for motion only because insome interval of time the moving body must traverse a distance, for Averroes itmeant the refutation of the vacuum.

The difference solution satisfied Aristotle’s requirement that motion is impos-sible if the two opposing tendencies are equal. It didn’t, however, maintain ratiosince the ratio of differences isn’t a simple constant. Arithmetic examples, simplynumbers pulled out of the air but usually simplified by employing multiples of two,argued against the Avempace difference solution because ratios are not preservedby differences. On the other hand, while the ratio solution explained the role ofthe medium in producing and maintaining motion it had a flaw. If the force equalsthe resistance, the velocity doesn’t vanish.

The theory of proportion is important for understanding both the Mertonianarguments and the later novelty of Galileo’s treatment of kinematics and dynamics.For “arithmetic proportion,” differences, the quantities being compared must beof the same quality, that is type; we would now say they must have the samedimensioned units. For instance, if we subtract two forces from each other, thedifference must be a force. But if this applies to the force law of Philoponusand Avempace, then the velocity cannot be simply proportional to the differencebetween the force and resistance. Instead, forces can be taken in ratio to forces,while velocities can be taken as ratios, and the resulting dimensionless numberscan then be compared. So to insist that Aristotle’s view meant a ratio of forceto resistance allowed a comparative ratio of velocities. The difficulty is this onlyvanishes in the limit of zero force, otherwise for any resistance—as Bradwardineand the Mertonians realized—any force no matter how small will produce motionwithout a threshold.

The continued application of a force maintains constant speed. To produceacceleration requires a change in the force and/or resistance. We can imagine,


then, a continually increasing force for a falling body as it nears its natural placeafter being released at some height. In any interval of time, the velocity increasesso the distance traversed in any interval is �s = 1

2 V�t where the factor of 1/2comes from the average speed during any interval �t . But since the velocityincreases as the time, this produces the law for continually accelerated motionthat the distance increases as the square of the time. Notice this doesn’t take intoaccount ether the source of the acceleration or the dimensionality of the physicalquantities involved. It simply translates into a quadratically increasing distancecovered and, therefore, a harmonic law for the intervals such that for successiveidentical intervals of time the increments of distance are a simple arithmetic series(1, 3, 5, · · ·). Both of these results would again appear in Galileo in the seventeenthcentury, while the quantitative aspect was developed both at Merton and Paris.

After dismissing these first two formal laws in the Tractatus, differences orsimple ratios, Bradwardine settles on one he describes as:

The proportion of the speeds of motion varies in accordance with the motive toresistive forces, and conversely. Or, to put it another way, which means the samething: The proportion of the proportions of the motive to resistive powers is equalto the proportion of their respective speeds of motion, and conversely. This is to beunderstood in the sense of geometric proportionality.

Since Bradwardine, in the first two books, had distinguished the types of propor-tions, the coda was precise for his readers: the ratio was taken in the form betweentwo, not three, quantities in the sense of force to force, and velocity to velocity.In the subsequent theorems, he elaborates on this with numerical examples thathave no physical basis. They’re just the usual simple arithmetic choices. Now inthe first sections of the work, Bradwardine presented proportions as arithmetic (ordifferences), geometric (the ratio of two quantities), or harmonic (a special case ofgeometric proportionality) following the propositions of Euclid, Book 5. Thus, ifwe take a proportion of proportions of the velocity we mean a number, N , raisedto the power of the velocity V such that:

N V = (F/R)N

for an arbitrary N ; this satisfies the requirement from Aristotle that if F = R,the velocity is zero. Interestingly, there is an anonymous abstract of the Tractatusde Proportionibus from the middle of the fourteenth century, not long after theTractatus, that provides a clearer presentation than Bradwardine himself (Clagett1967):

The fifth opinion, which is the true one, posits that the velocity of movement followsthe geometric proportion of the power of the motor above the power of the thingmoved. Whence the meaning of that opinion is this: if the two powers and the tworesistances and the proportion between the first power and its resistance is greaterthan the proportion of the second power and its resistance, the first power will bemoved more rapidly than the the second with its, just as one proportion is greaterthan the other.


which can be rendered, symbolically, as

F2

R2=

(F1

R1

)V2/V1

thus making clear the proportion of proportions part of the formalism and thatwe are not here comparing incommensurate things. The formula derives from theproportion of proportions since, if we have two quantities A : B and B : C, thenA : C = (A : B)2, found by dividing both sides by C. Where does this force lawcome from? Certainly not from any test in our understanding of empiricism. Itsatisfies all the requirements of dynamics and that’s enough.

Geometric reasoning was particularly important to the treatments of this prob-lem and it spawned a new style of reasoning. Mainly founded in the theory ofproportion, by equating number (quantity) to attribute (quality), figures—a sort ofgraphical solution—was developed for representing accelerated motion. It seemsto have originated with Nicole Oresme (born around 1323, died 1382) and Gio-vanni di Casali in the early fourteenth century. Again, this is used to illustratethe text, not as a calculational tool. Uniform quantities are those which can berepresented by rectangles, those for which the intensity does not change withextension. A non-uniform quantity, called uniformly difform, is represented by aright triangle, and as Oresme states in On Configurations of Qualities,“a difformdifform (non-uniformly increasing quantity) is a combination of diagrams. A qual-ity uniformly difform is one in which, when any three points [of extension] aretaken, that proportion of the distance between the first and second to the distancebetween the second and third is in the same proportion as the excess in inten-sion of the first over the second to the excess of the second over the third.” Hecontinues, “Every measurable thing except number is conceived in the mannerof continuous quantity, hence it is necessary for the measure of such a thing toimagine points, lines, and surfaces—or their properties—in which, as Aristotlewishes, measure or proportion is originally found. Here every intension which canbe acquired successively is to be imagined by means of a straight line erectedperpendicularly on some point or points of the [extendible] space or subject of theintensible thing” (Clagett 1959).

To create a graphical representation, the quality is taken as the vertical axis andthe quantity is taken as the horizontal. In other words, a velocity is a quality, thetime is the quantity. Then the mean value theorem follows from taking the meanquality (motus) as Vfinal/2 and the quantity, time, as �t so if the distance coveredin an interval is V�t it follows that the distance covered varies as (�t )2. Thusa constant quality is pictured as a quadrilateral and every quality that changesuniformly, uniform difform starting from zero is represented by a right triangle.Every quality that is uniformly difform starting from any arbitrary initial value isa rectangle surmounted by a right triangle. This is a general method that Oresmethen applied to motion. Every velocity occurs in an interval of time the durationof which is the base of the figure, or the longitude. The quantity or intension ofvelocity is the vertical axis or latitude. So figures can be geometrically compared,since they are similar, as ratios of similar quantities (time to time, velocity to


velocity), and the distances covered are then also formed from ratios (since thedistance is the area of the figure). Even if we are comparing quantities that arenot the same (two velocities relative to two times) the graphical construction madeit simple to imagine how to do this. This kinematic construction isn’t directlyrelated to forces, but it shows how a mathematical scheme was developed in thefourteenth century that could handle accelerations. The rule of proportions wasstrictly followed, motions were compared to motions, and intervals of space to eachother, and intervals of time to each other, all separately. No compound proportions,such as products of velocity and time, were used so this approach doesn’t requiredimensional specification. Each quality is separately considered.

Uniform velocity, required a different explanation than an accelerated quality.But once the mathematical machinery was in place for describing one type ofacceleration, it could deal with any. Swineshead and Oresme knew this well,they both asserted that any variation of accelerated motions can be represented byalterations of the geometric figure. In this, dealing with rectilinear motion, they alsoknow that accelerations are simply additive. The cause of the acceleration was notimmediately important, that would come with the Italian mechanicians about twocenturies later. This method can be easily extended to include any representation ofthe change in the magnitude so the Merton rule follows immediately: in computingthe interval covered use the mean value. So a speed that is uniformly changedduring an interval is the same (in the later language, as a constant acceleration) sothe distance covered in any interval of time—starting from the motion—increasesas the square of the time. This is a precise kinematical definition, nothing more.It lacks any explicit statement of the cause of the acceleration. But, since theunderlying assumption is that no motion occurs without a driver, and any changein the motion reflects a change in the driving, the dynamical implication is alreadybuilt into the discussion. As long as motion cannot occur except by continuingeffort of agents, internal and/or external, that change the qualities, any kinematicdiscussion is automatically a dynamical treatment. Without inertial motion, theseparate treatments were unnecessary. Any change in the driving that produces avariable acceleration can be broken into intervals of constant acceleration. Yousee already the need to consider infinitesimal intervals of change. Note that thisconstruction is extremely general, any property can be graphed that depends onany interval, it isn’t restricted to motion—for example, the “‘hotness” of a bodydepending on the amount of fire in which it is in contact or its extension withrespect to an applied weight—so this is a tentative start on quantification. It justwasn’t used that way at first.

On the issue of intension and remission of forms, one good example comesfrom an anonymous piece from the Venice 1505 edition of the Tractatus, entitledSummulus de Moti Incerti Auctoris:

4. It is noteworthy that there are two kinds of intension, quantitative and qualitative.Quantitative intension takes place by adding quantity to the maximum degree of thatquantity. Qualitative intension can take place through rarefaction or condensation,e.g. if there were a body one half of which was at a maximum hotness and in theother part the heat was remitted. Then if the more remiss (i.e. less hot) half werecondensed, while the other part remained as before, the whole is understood to be


hotter because the ratio of the more intense to the more remiss part will be continuallyin greater proportion . . . And just as space is that which is to be acquired by localmotion, so latitude is that which is acquired by motion of alteration. And in the sameway that a moving body which describes more space in an equal time is moved moreswiftly as to the space, so that which by alteration acquires more latitude in a certaintime is altered more swiftly.

With 20–20 hindsight, it seems astonishing that the Scholastics made virtuallyno appeal to experience when proposing these force laws. Some of the simplestdemonstrations, such as comparing dropped bodies or looking at trajectories ofprojectiles, were never reported in the mechanical works. There were a few re-markable exceptions, the English thirteenth century philosophers Grosseteste andRoger Bacon, but their interests were limited mainly to optical and meteoro-logical phenomena and even those were not reported in a way that makes theirmethods clear. The general procedure was to propose a law through the filter oftheir presuppositions and use only logic to arrive at a conclusion. In this man-ner the commentaries were frequently like the illustrations in the manuscripts,reproducing rather than extending the text. Even in astronomical texts, the phe-nomena were rarely represented from nature, even when tables are included thedata are often merely geometrically corrected for the appropriate epoch of the textand not from new observations. The diagrams were intended to guide the readerand, although this wasn’t the rule, occasionally drawn by scribes who had lit-tle understanding of the material they were copying. When comments are madeabout everyday experiences they are of a very general sort. The Scholastics, ingeneral, distrusted the senses and lacking empirical principles to guide themeschewed experiment as a means of discovery. The texts themselves were theobjects of study and the commentary was the principal tool for explicating thetext, citing authorities to expand or contradict the material. But this was about tochange.

THE SETTING OF RENAISSANCE PHYSICS

On 23 February 1455, Johannes Gutenberg completed the printing of the “Bible in42 lines,” the first book produced in Europe using movable type. Within 50 years,printers had begun publishing texts in Latin, Greek, and Hebrew throughoutEurope with astounding results. At last texts that could only be painstakinglyreproduced by professional scribes in single copies were available in tens, oreven hundreds, of copies. The efficiency and accuracy of the manufacture, and itscomparatively low cost, along with the expansion of centers of paper production,made it possible for the literati in very different, distant places to read exactly thesame text. As the reading public expanded so did the demand for new material andthe rate of book production rapidly grew along with the range of topics. It shouldn’tbe thought that producing a book wasn’t a difficult, time consuming job, but in thetime it took to copy a single manuscript it was possible to obtain dozens of “pulls”of the pages from the press. Not only classics and commentaries issued fromthe new publishing firms. Poetry, romances, music, histories, maps, and—mostimportant for our discussion—scientific and philosophical works began to appear.


The loci of activity expanded. New schools and universities were being foundedbut there were also the secular courts of the powerful families, for whom a localsavant was a prized “possession” and adornment; perhaps the most obvious exam-ples are the Dukes of Urbino and Ferrara, the Medici of Florence, and Rudolf IIof the Hapsburgs. Cities, such as Venice, and royal patrons continued their roleof supporting this new intellectual class along with an increasing interest of theecclesiastical authorities in the exploding scholarly activity (though rarely for thesame reasons).

A new figure, the knowledgeable amateur, appeared on the scene. Leonardo daVinci (who we will discuss presently) is certainly the prime exemplar but therewere other artists, such as Albrecht Durer, Botticelli, Michelangelo, and Raphaelwho were fascinated with the new ideas of nature and the abstractions of spaceand time. But there were also those who had the wealth and leisure to engage inthese activities full-time, Tycho Brahe and Ferdinand II of the Medici family areamong the most famous. These all now had access to a wider dissemination oftheir ideas through the press and in some cases, such as Tycho’s, their works werefundamental influences in the debates of the scholars.

An increasingly complex set of scientific instruments also became available inthis period, produced by skilled specialists. More advanced glassblowing tech-niques, for instance, not only benefited alchemical studies but also physics, thebarometer and thermometer are examples. Clockwork and gearing, and astronom-ical instruments (not only the telescope at the start of the seventeenth centurybut also astrolabes, quadrants, compasses, and geometric instruments) improvedthe accuracy of measurements. On a darker side, the introduction of gunpowderand the increasing use of ballistics in warfare required increasingly more preciseknowledge of dynamics and materials. An archer required only skill to be effective,having complete control of the tension and aiming of the arrow and bow. To hit adistant target taking full advantage of the capabilities of the new artillery requiredincreasingly precise methods for computing trajectories and the right quantity ofexplosive propellant. Rules of thumb went only so far, understanding their originsbecame a major theoretical occupation that continued for many centuries (evennow). These developments were not only in the area of dynamics but also statics,since fortifications needed to withstand new perils. The doctrine of impetus becameincreasingly useless for computing quantitatively ballistic motion and, as we willsee, even called into question some of the foundational assumptions of physics.

The great architectural projects of the Renaissance, inspired by the classicalworks of Vitruvius and Hero, provoked a new interest in the abstract study ofphysical behavior of materials. Grand projects are costly and the exigence of thetime was originality. It wasn’t enough for a master builder to slightly modify anexisting, successful design by scaling it up or down to the demands of a newcontract. The enormity of some of the new structures, for instance Brunelleschi’sdome for the Santa Maria del Fiori cathedral of Florence and Michelangelo’s andDella Porta’s for St. Peter’s basilica in Rome, required an advanced understandingof materials and stresses. While many of the studies, those of Leonardo being themost detailed surviving examples, were mainly qualitative, the phenomena werebeing carefully examined with an eye toward the underlying principles and causes.This was the environment in which science entered a new phase, in which the


empirical and theoretical, joined in the language of mathematics, produced a newphysics.

EXPERIENCE AS GUIDE: LEONARDO DA VINCI

Leonardo (1452–1519) was a bridge between two epochs in many aspects ofwestern culture. His contemporary reputation rested on a comparatively limited—but remarkable—artistic production, a few disciples of outstanding quality, andhis notable technical expertise. Our usual categories, however, fail properly todescribe his activities. He was an itinerant artist working in the employ of sev-eral epochal patrons in Italy and, eventually, in the French royal court. He wasan engineer—really a technical consultant—to the Sforzas (Milan) and Medici(Florence). Although none of his grandiose projects were ultimately realized, hedid have a significant influence on later developments, especially in fortificationand urban planning. His reputation as an empiricist and precursor of develop-ments in the physical and biological sciences comes from scholarly studies thathave mined his scattered literary productions since the late eighteenth century.

Leonardo is important for our discussion of physical ideas in the late middleages precisely because his results were arrived at and preserved in such privatecircumstances. His notebooks record the musings of an amateur. He held neitheran academic nor court position, nor was he a theologian or natural philosopher.If we are to judge him by his own standards, we can use a letter to Sforza, theDuke of Milan, as his “curriculum vitae” for possible employment in the serviceof the Duke. He writes of his skills as an architect and practical mechanician,expert in military construction and ballistics, hydraulics, and urban planning.What we find in his notebooks, in contrast to this presentation, is a brilliantlyobservant, intensely curious, outstandingly empirical individual who frequently,and with erratic motivations and methods, addresses often profound problems witha very personal mythology and perspective. It is precisely because some of thesemusings, often queries without significant elaboration, but also long connectedfragmentary gropings along the way to the resolution of a particular question, thatLeonardo has often been presented as a modern. This is often the case in his na-ture studies, which have the stamp of a biologist and systematist. But Leonardo’sphysics show that he was an Aristotelian in his theoretical dynamics, even thoughhe was often inconsistent in his views. He was above all else an applied scientistwho established the concept of copying from nature by abstraction. In this, heachieved what Bacon and Grossteste had advocated to a less sympathetic audi-ence three centuries earlier: that experimental science and observation can testknowledge and correct first principles. His work shows an acquaintance withmathematics but not with formal geometry, so his ideas often lack the precisionArchimedes’ analysis. Nevertheless, Leonardo was able to extract some fundamen-tal concepts from relatively simple observations, often of things in their natural set-tings. For example, his study On the Flight of Birds represents a founding work inbiomechanics and, consequently, in the application of the force concept to theanalysis of living things. Here Leonardo’s insights far outstripped his abilities toformalize and extend them. His scaling arguments are based on simple ideas of liftnot associated with the movement of a body through air but just on an extension


of the support by atmospheric pressure. His studies of turbulence are outstandingand are relevant to our discussion because they provide the first systematic anal-ysis of the motions of a continuous medium. Descartes used remarkably similarideas in his theory of planetary motion without any knowledge of his forbearer.

Leonardo’s extensive studies of statics bear greater similarity those of Heroof Alexandria than Archimedes. His aims were generally practical although hisobservations often go beyond experience to attempt to extract general principles. Inhis description of weights suspended by cords of unequal lengths, he qualitativelydescribed the law of addition of forces. How does a force alter a body? How does itachieve balance when there are different internal and environmental influences?Questions of this sort were particularly compelling for Leonardo and to addressthem he extended the mechanistic tradition of Archimedes and Hero. For practicalpurposes, he was also especially interested in how materials behave under stressingand loading. His discussion of positional gravity was along the lines pioneered byJordanus but when it came to extended bodies he went farther. His description ofthe differential reaction of a stressed beam displays his originality and illustrateshis way of reasoning about mechanical problems:

If a straight spring is bent, it is necessary that its convex part become thinner andits concave part thicker. This modification is pyramidal and consequently here willnever be a change in the middle of the spring. You shall discover, if you considerall the aforementioned modifications, that by taking part ab in the middle of itslength and then bending the spring in the way that two parallel lines ab touch at thebottom the distance between the parallel lines has grown as much at the top as ithas diminished at the bottom. Therefore, the center of its height has become muchlike a balance for the two sides. (Codex Madrid, trans. Carlo Zammattio)

With this remarkable statement, Leonardo empirically anticipated a fundamentalaspect of deformation that appears to have been ignored by all earlier Westernwriters and would not appear again until the eighteenth century. I’m emphasizingthis as a striking example of how relatively simple the reasoning could be whenphysical problems were stated geometrically and how notable it is that nobodynoticed something that Leonardo considered almost obvious. We will return tothis in more detail when we discuss the developments in statics after Newton.4 ButLeonardo’s dynamics was, in contrast, consistently expressed in the framework ofthe impetus description for and explanation of ballistic motion. He never system-atized his observations in geometry, but his jottings show that he viewed “violent”motion as a continuous variation between a curved trajectory and freefall. Theflight of a cannonball was, for instance, a continuous curve that gradually lostimpetus and was replaced by natural motion in an asymmetric arc.

NOTES

1. The Method was only recovered at the very end of the nineteenth century whenHeiberg, one of the foremost scholars of ancient Greek mathematics, recognized thepresence of the text under another theological text, virtually scrapped of the parchment toreuse the precious material. Its final recovery, transcription, and publication came morethan 2000 years after its composition. Such was the fate of many of the works of the Greeks


and Romans for which we know only the titles and abstracts of the contents from latercommentators, and not only in Europe.

2. It is rare that history can be re-lived as well as recovered but there is actually aninteresting analogy to the Medieval reencounter with the ancient texts taking place in realtime. The medieval city Timbuktu, now in Mali, was one of the principal trading posts alongthe Niger in the twelfth century. The cache of manuscript material, randomly collectedthere, has yet to be explored. While it is unlikely that any great number of previouslyunknown texts will be discovered amid the manuscript material, it is something you canwatch develop with its historical precedent in mind.

3. I think an aside is in order here since so much of the technique used in the fourteenthcentury discussions has been preserved in the pedagogy of modern introductory physicstextbooks. To a modern reader, so accustomed to nearly instantaneous communication,mere chronology doesn’t, I think, make clear how hard it was to pass from the subtle disputesof the Schoolmen to the first dynamical studies of Tartaglia and his Italian contemporaries.Keep in mind, almost 200 years separated the major works of the Mertonian and Parisianschools and the Northern Italians. And it would take nearly 60 years after that before theconcept of force would be harmonized with these kinematic, or semi-kinematic, treatises.

4. Strings, and springs can be treated in one dimension, at least initially. But whenyou have a beam, or a membrane, or a plate, the problem is compounded by the multidi-mensional nature, and multidirectional character, of the surface and forces acting on it.For example, a string can be described by a linear relationship between the applied stress(force, loading, weight) and the reaction strain: it changes in length and the more stressis applied the greater the distension. This is the first form of the law described by Hooke(1672) and Mariotte (1680) for the linear dependence of the deformation (strain) to theapplied stress. The innovative point is that the relation is a simple proportion, with theconstant of proportionality being a specific only to the material and not to the deformation.This linear law, the truth of which is restricted to a limited range of stresses and materials,was a step far beyond the rheology available to Galileo.


3

THE NEW PHYSICS

Let sea-discoverers to new worlds have gone,Let Maps to other, worlds on worlds have shown,Let us possess one world, each hath one, and is one.

—John Donne, The Good-morrow

The span of two centuries between c.1500 and c.1700 marks the period usuallycalled the Scientific Revolution, between the appearance of the heliocentric systemof Copernicus to the age of Newton, neither is a firm endpoint. As we’ve seen,Aristotelianism was deeply rooted in the cultural soil and held firm for a verylong time, throughout this period. But at the same time there was a new, growingsense of the importance of actual observation and experimentation as a way todiscover regularities of nature. Gradually the intellectual climate shifted from onedepending on authority, the more antique the better, to individual experience.Instead of being content with repeating, commenting on, or criticizing ancientauthorities, it was an age of hypotheses. It opened with an epochal event: the firstEuropean encounter with the Americas. It was a disruptive shock. Suddenly thephenomenology of the world was populated with peoples and species of plantsand animals never before seen. The categories no longer held so simply as theyhad in the Old World, and the exploitation of these new discoveries requirednew ideas and methods. As warfare had transformed during the previous century,now commerce changed. The world wasn’t simply a sphere, that was already wellknown of course long before Columbus and the voyagers. It was far larger, far morediverse, and much less familiar after the beginning of the sixteenth century. It wasan age of discovery.

HELIOCENTRICITY: COPERNICUS

The separation between cosmology, that is the physical explanation of the planetarymotions, and astronomy, which dealt mainly with their description (and astrology,


Figure 3.1: The illustration from de Revolutionibus showingthe explanation for the seasons related to the constancy ofthe direction in space of the rotational axis of the Earthduring its annual orbit. Image copyright History of ScienceCollections, University of Oklahoma Libraries.

treating their interpretation), allowed fora broad latitude in any mathematicalkinematics. It also produced strikingcontradictions. We’ve already seen thatepicyclic motion violated at least oneprincipal that any body should have asingle natural motion. In Ptolemy’s sys-tem, with both the epicycle and theequant display compound natural mo-tion about centers other than the Earth.Yet the system was made at least min-imally consistent by imposing uniformmotion of the deferent around the cen-ter of the world, even though it was ec-centric. In contrast, the Pythagorians as-serted that the Earth moves around thecentral fire of the Sun, though withoutdeveloping either a full scale physicalpicture or a predictive geometric appa-ratus. The system survived as an almostmystical tradition for a very long time,even long after the critiques by Aristo-tle, and was best known through thosecitations since the adherents deignedto write anything substantial. The mostdramatic departures from Aristoteliancosmology came not from the physicalbut the mathematical side, the hypoth-esis by Aristarchus of a moving Earth.None of the physical consequences ofthis hypothesis were examined, as faras we know, but its evident simplifica-tion of the motions exerted an enduringinfluence. This is especially interestingbecause, aside from his work on the dis-

tance to the Sun, Aristarchus’ cosmological construction opinions were only knownthrough a brief mention in Archimedes’ work the Sand Reckoner. For our purposes,it represented more of a model representation of, than a physical precursor to, he-liocentricity.

If we reserve the term “solar system” for the complete model that combinesthe physical and mathematical principles in a self-consistent way, we have towait nearly two more centuries, but the process began dramatically with NicholasCopernicus (1473–1543). From his first sketch of the idea, the Commentariolis of1513, to the publication of his crowning achievement, de Revolutionibus OrbiumCoelestium, took 30 years and he never lived to see its impact. The book’s principalarguments were popularized by his pupil, Georg Rheticus (1514–1576), who wrotea summary, the Narratio Prima, that circulated separately as sort of a lecture on

The New Physics 41

the main ideas. What Copernicus had done was more than merely making the Earthmove. He saw what this implied in the big picture, that previously diverse andunexplained phenomena could be included naturally within the world system andthat, in consequence, new physics would be needed to account for the motions.But he did not live long enough, nor have the necessary physical concepts athand, to take that last step. For Copernicus, celestial dynamics, the mechanicalpart of the work, remained largely unaltered from those of the Hellenistic systemof the spheres. A driver beyond the system, no matter how much enlarged, wasstill required and the action was transmitted to the planets by the spheres. Whatchanged was that this outer driver didn’t need to participate in the motion ofthe plants. The rotation of the Earth, not of the eighth sphere, explained all ofthe motions of the fixed stars, whether daily or the precession of the equinoxes,by “reflection.” The Moon moved with the Earth, somehow. These were morethan hypotheses of convenience, for Copernicus they were real. But although thecenter of this world had been displaced from the Earth to the Sun, there was littlechange in the construction by the humanist scholar Georg von Peurbach almost 70years before in his Theorica Nova Planetarum, in which the planets ride like ballbearings between two rigid spheres in what looks like a mechanical transmission.But for von Peurbach and his contemporaries, except Copernicus, each moverrequired a different construction centered separately on the Earth. This is thesweeping innovation of de Revolutionibus: the system was unified and it avoidedmany of the paradoxes of the geocentric system, such as mutually intersectingepicycles and unphysical changes in the angular diameters of the Sun and Moon.To account for deviations in the orbital rates, planets were carried on smallepicycles, and the effect of the equant was replaced by additional small epicyclesto maintain the constancy of local circular motion (that is, motion was alwaysassumed to be constant around the center of the path, eliminating both the eccentricand the equant). The truly remarkable elements of Copernican heliocentrismwere the requirement of the multiple motions of the Earth and Moon and theconception of a force not centered on the Earth. Although eschewing any explicitdiscussion of dynamics, it is clear that Copernicus’ views require a fundamentalreassessment of Aristotelian physics, so much so that Andreas Osiander, whoedited the first edition and guided the work through the press, felt compelledto add an anonymous preface in which the system is called only a “hypothesis”from which no physical implications should be drawn.1 That Copernicus himselfunderstood those implications is, however, not in doubt. Following the sequenceof the arguments from the Almagest, Copernicus presented in his first book adetailed description of the origin of the seasons—due to the immobility of theEarth’s rotational poles, and a mechanism producing precession of the equinoxes.The annual motion and fixity of the poles is very important for the force concept.It implicitly uses, for the first time, what would later be called conservation ofangular momentum, although this idea isn’t found in de Revolutionibus. The other,fundamental, feature of the system is that not only were bodies endowed withmultiple natural motions, but one of them is the rotation of the Earth.

Epicyclic motion is a produce of the frame of the observer and is, in effect, achoice. If viewed in the stationary frame centered on the Sun, planetary motionis always direct and periodic. In a rigidly rotating system, one for which all


bodies have the same orbital period, all frames are kinematically equivalent. Butif the periods change with distance, motion must reverse for some of the bodiesdepending on your choice of point of observation (that is, if you are on one of themoving bodies). This was understood by Copernicus but most clearly explainedby Galileo (in the Dialogs on the Two Chief World Systems, almost a century later)and exploited by Kepler (in his Epitome of Copernican Astronomy). The stationarypoints result from those intervals when the planets pass each other and, from theviewpoint of the more rapidly moving body, the slower is overtaken. Then theconfinement of the motion to closed orbits produces the strict periodicity in thesewith the separation of the times of the event being dictated by the difference inthe frequencies of the two orbits.

In a broad sense Copernican kinematics was simpler than classical modelsbut not entirely free of its constructions. To account for the irregularities in theperiodic motions (the principal anomaly), Copernicus still required small epicyclesthat would produce the necessary deviations in the rates. Remember, there wereno absolute distance scales so there was still no way to connect the motions withterrestrial forces. The motion of the inner planets, which we can now properly namebecause of the change in the reference frame, found an immediate explanationwith the heliocentric construction: Mercury and Venus are slaved to the solarmotion because of their relative distances from the Sun and their location interiorto the Earth’s orbit. The relative distance scale, that was lacking in the Almagestand all subsequent work, was introduced, at least for the inner planets. Usingpurely geometric arguments, the sizes of the orbits relative to the Earth’s couldbe determined since this depends only on their maximum angular separation fromthe Sun.

A mobile Earth removed the greatest cosmological impediment remaining fromthe Aristotelian system. Now the world could be infinite, the stars were no longerfixed to a sphere that transmitted the force of the Prime Mover to the planetarymachinery. A new source was needed for the motions.

In the dynamics of the spheres, since the construction was designed to “preservethe appearances,” each motion was identified and treated separately. The meanmotion that defined the circular path around the ecliptic was a reference motionbut each departure from that referent was treated separately using the same ora related mechanism. The complication of the model was irrelevant, its successwas enough to guarantee its adoption. All the various bells and whistles tackedon to the fundamental construction, the eccentric, equant, moving equant, and thelike, were all required alterations whose physical origins were deemed beyond thescope of the approach. The simple necessity of accounting for a departure from asuccessively more complex reference motion was the product of necessity and themathematical tractability of the picture was enough to recommend the procedure.

For Copernicus this was still a valid method. With the displacement of the centerof motion from the Earth to the Sun came new questions but not a real change inthe physics underlying the machine. It was still a machine in the sense that itsparts were truly driving mechanisms that maintained the structure and kinematics.It may now be hard to imagine, at a distance of so many centuries, how effectivelythis separation could be maintained. Even in the face of mounting contradictionsin the hypotheses regarding the dynamics, the calculations of planetary positions

The New Physics 43

could be affected without ever asking such questions. Those who chose to avoidasking cosmological questions were able to do so without having to ally themselvesto a physical explanation of the system. For Tycho and those who attempted tomaintain some point of contact between the physical causes and the observedeffects, this was the weak point of the Copernican approach but nonetheless itwas still a matter of almost personal preference which model one chose for ageometrical arrangement of the principals.

The physical contradictions were there in abundance, however, and even Coper-nicus was aware of some of the consequences of this altered construction. He never,however, addressed them in a systematic way, focusing instead on the advantagesof the new ordering. For Rheticus this was also acceptable, as it was for otherCopernicans. Actually, the theological and philosophical consequences of thenew system were far more disturbing than their physical implications. The motionof the Earth was, perhaps, the one sticking point. The arguments adduced byPtolemy and in the Middle Ages for the stationarity of the Earth were so obviousand simple that they sufficed to persuade most casual critics that the system couldfunction as a simple change in computational reference frame but had no othersignificance. For the Copernicans, the physical reality of the system—as opposedto its mathematical convenience or efficiency—was more dramatic. It must bepossible to decide by experience if the Earth is really moving or not. If it isn’t,the heavens are in one way far simpler, the universe is a closed thing and Godcan be in His heaven. If, on the other side, the Earth is mobile, then this wholeconstruction fails to stay together in a unified sense. You could say it was thehappy mistake that the clerics of the Renaissance chose to base their theology onphysical demonstration rather than simply using faith and morality. But they sochose and this opened the need to keep the whole coherent.

EXPERIMENT AND OBSERVATION: WILLIAM GILBERTAND TYCHO BRAHE

In 1600, William Gilbert (1544–1603) authored the first systematic experimentalanalysis of a force. In de Magnete (“On the Magnetic”), he presented detailedaccounts of an enormous variety of experimental studies he conducted on thelodestone, the naturally occurring iron mineral material from which magnet iswere originally made. As a physician—he was the court attendant to Elizabeth I—Gilbert was schooled in Galenism with its language of “humors,” “efluences,” and“affections” and he used this language to describes the phenomena he observed.In the work, he never mention force, or even impetus, but his affinities andaffections—the same terms used by the alchemists of the period—are similar inintent if not in detail.

Gilbert’s method was purely empirical, drawing his “hypotheses” solely fromexperiment. He was the first use a material model as a representation of a physicalbody, in this case leading to his principal contribution: the realization that theEarth is a magnetic with its source within the body. This allowed him to use acompass needle, placed alongside and moved around a spherical lodestone, tosimulate the behavior of the then common mariner’s instrument. His explanationsof the dip (declination) and orientation (deviation) of the needle as a function of


Figure 3.2: A figure from Gilbert’s De Magnete show-ing the “influence” of the magnetism of a lodestonein the form of a sphere, the dip of a magnetic nee-dle. This model of the Earth’s magnetic field, the ter-rela, provided the empirical basis for later attempts inthe seventeenth century (notably by Edmund Halley)to determine longitude with magnetic measurements.Image copyright History of Science Collections, Uni-versity of Oklahoma Libraries.

longitude and latitude provided the theoret-ical basis for navigation and, later in thework of Edmund Halley, produced the firstmeans for determining longitude indepen-dent of celestial measurements.

Gilbert was a forceful advocate for theCopernican system. In the last book of deMagnete he presented defense of the helio-centric system but went even farther intothe physical consequences than had Coper-nicus. He takes two enormous leaps, bothextensions of his magnetic doctrine. One isthe banishment of the spheres. Tycho Brahehad done the same in his modified geocen-tric system. In their place, Gilbet assertedthat the Sun is the source for the motionof the Earth and the planets through a sortof magnetic influence. The same idea wouldbe proposed, in a more physical way, by Ke-pler. Gravity is something distinct and dif-ferent for Gilbert as for all others, the actionof a magnetic is something almost “volun-tary” on the part of the substance and not allobjects feel the influence of the lodestone.The other is a much more wide-ranging idea,that the stars are not only distant but at avast range of distances, and that there areinnumerably many fainter than the limitsof vision. While he leaves this unexploredand draws no other implication from it, forour purposes this hypothesis is a suggestionthat the open universe, which Newton wouldneed for his system, was already beingconsidered a century before the Principiaand a decade before the Galileo announcedits reality in the Sidereus Nuncius. In the

final analysis, Gilbert’s book represents a bridge rather than a starting point. Hisexperiments were reported in almost mystical terms.

Gilbert wasn’t alone in his phenomenological approach. A fellow traveler alongthis empirical road, the Danish nobleman Tycho Brahe (1546–1601), proposeda significantly modified—but still essentially Aristotelian—system of the worldat the end of the sixteenth century. His attempt was to reconcile two contradic-tory exigencies. One arose from his astronomical observations, especially of thecomet of 1577. Finding that this body was superlunary, in showing no detectableparallax, Tycho’s construction of its orbit definitively required that it cross theboundaries of the spheres. Thus, absent now the principal driving mechanismand the basis for the previous unification of the planetary motions, he proposed a

The New Physics 45

radical non-Copernican alternative to the Ptolemaic conception, almost 50 yearsafter the appearance of de Revolutionibus. Tycho’s systematic observations of plan-etary motion and stellar positions, and his determination of the motion of cometsand their implication for the driving of the planets, were important for the re-form of astronomy but ultimately secondary to the development of physical theory.Like Gilbert and others of the sixteenth century, he was content to “preserve theappearances” without mechanism. His planetary theory is a mathematical con-struction, still couched in vaguely Aristotelian terms—the immobility of the Earthis assumed—but with a bizarre arrangement of the planetary order to account forthe motions. The Earth is immobile in this system, the planets circulate around theSun, but the Sun moves around the Earth along with—and above—the Moon. Hissystem is purely kinematic once the motion is constrained to remain geocentric.It can be nothing else since he dispensed with the driving spheres and, lackingany alternative explanation for the motions, was forced to present the system as amathematical hypothesis without other justification. Although it was sufficientlynovel to attract some adherents. Even after Galileo (as late as the mid-seventeenthcentury), Riccioli and his fellow Jesuits found it more acceptable than the Coper-nican alternative since it was capable of producing positional predictions while notconflicting with the theologically proscribed geocentricity. But for our narrative itremained just a sidelight.

Gilbert and Tycho represent transitional figures in the history of physical sci-ence. Both recognized the limitations of their predecessor’s methods and conclu-sions, and they were convinced that quantitative empiricism was the only wayto study nature, but neither was able to break with the principles they inheritedfrom their medieval roots. At the same moment Tycho demolished the crystallinespheres, he sought to maintain a geocentric world and Aristotelian dynamics. Evenas Gilbert discovered the properties of magnetic and electrical substances withsuperb experiments, his speculations invoked Galenic mechanisms of humors andinfluences. Tycho could maintain a purely geometric construction that violatedeven his physical axioms and swing between descriptions of the planetary motionsand terrestrial phenomena without a hint of admitting their basic contradictions,maintaining an implicitly agnostic stance in lieu of a unified system. Gilbertcould create specially tailored forces to account for phenomena without askingabout their more general applications. Thus, at the opening of the seventeenthcentury, although the earlier worldview was showing fatigue, it had yet to crack.That required a new approach, a new openness, and a greater willingness to trustexperiments and mathematical reasoning.

KEPLER: ASTRONOMICAL PHYSICS ANDNEW PRINCIPLES

The Copernican system, as a mathematical construction, was more than tolerated.It was argued, it formed the intellectual fodder for countless debates and contrastsbetween calculational systems for nearly two centuries. But the question of whichmethod—given that by the start of the seventeenth century there were threeavailable—provided the best means for computing planetary phenomena is notwhat we’re concerned with here. If we are talking about physics, the picture is


quite different. Many of the debates were cosmological—that is, philosophical ormetaphysical—and they centered on the mechanism(s) implied by the choice ofsystem. What basic physical picture Copernicus pictured, we can only hypothesize.But he had made a spectacular leap of imagination that would drive many morein the centuries ahead, one that is much different than just a mathematical choiceof reference frame: without justification, raised to the level of a postulate, themotion of the Earth is used to examine all observable phenomena of planetarymotion independent of explanation. To drive the system may have required asystem of spheres as complex as the medieval cosmos, but that’s not important. Byavoiding explicit reference to the origin of motion, and instead concentrating onits manifestations, Copernicus had evaded the most serious criticism that wouldhave diminished or even removed any credibility in his system. In eschewingdynamical hypotheses, the construction could stand on its own and, if successfulin reproducing the motions of the planets, inspire research programs to deal with itsphysical origin. Nothing in de Revolutionibus , however, leaves any doubt about thenature of the Copernican program. The most direct statement is contained in theCommentariolis, a short work from around 1514 that provides a precis of the largerresearch project. There the assumptions are written in unelaborated style and setout at the start of the long work that ultimately produced Copernicus’ monument.Whatever philosophical, theological, or personal motivations Copernicus may havehad for placing the Sun as the immobile center of the system are unimportant. Hehad changed the construction of the world and this required a new mechanism.That was Kepler’s program, to complete the solar system.

We encounter in this new science a new personality, the mathematical physicist,and none is a better example than Johannes Kepler (1571–1630). As an astronomerand mathematician, and above all a Copernican, he took the physical speculationsof Gilbert and a remarkable knowledge of observational astronomy from Tychoand forged a new way of unifying physics, a celestial mechanics. This he achieveddespite having initially been an adherent of Tycho’s system, and the inheritor ofTycho’s observational data compilation of planetary motion. Kepler was the firstto demonstrate that the Copernican world system actually leads to new laws, histhree laws of planetary motion. The result could not have been obtained withoutadmitting the motion of the Earth. Copernicus knew that a change in the referencesystem from the Earth to Sun centered coordinates removed a temporal alias in theplanetary positions, the effect of the annual motion of the Earth. Thus the periodsof the planets were now systematically increasing with distance from the Sun.He also knew that he no longer required the extraordinarily complex, multiplemotions for the planets, the necessity for the equant vanished, but he retained asmall geometric crutch, a sort of small epicycle, to produce the proper variationin the angular motions. This was only a mathematical device lacking any furtherrefinement.

What Kepler found was a set of geometric and dynamical laws. The first is thatthe areas swept out by a line extending between the planet and the sun are equal forequal intervals of time. Another way of stating this is when the planet is closer tothe Sun it moves more rapidly. This held not only for the projected angular motionsas they had been for all previous theories, the physical speed varies depending onthe distance; the changes are not merely uniform circular motion viewed from a

The New Physics 47

different vantage point. Thus, if r is the radius line and v is the angular speed(that is at an moment tangent to the orbit), rv is a constant. The second is thatthe planets orbits are the closed conic sections, circles and ellipse, with the Sunremaining immobile at one focus and controlling the motion, the opposite focus isempty. Kepler overcame his initial reluctance to accept a noncircular path onlybecause of what he saw as the overwhelming observational constraints derivedfrom Tycho’s data. The last, which is called the Harmonic Law, is that the squareof the orbital period of a planet increase as the cube of its mean distance fromthe Sun. The statement of this law is actually an implicit proportion since theunits Kepler employed, years and the astronomical unit—the mean Earth–Sundistance—actually scale all orbits to the Earth’s. Note that all of these laws arethe products of the change in reference frame: without the mobile Earth, there isno basis for assigning distance nor is the periodicity understandable.

Kepler’s dynamical explanation for the laws is uniquely his: the Sun, as thecore of the planetary system, is also the source of its motion. But it is at thispoint that Kepler’s dynamics echo some of the earlier doctrines. Orbital motionis instantaneously tangential. To continue, according to the construction in theEpitome and Astronomia Nova, this motion is always driven. Having no spheresto serve this role, Kepler located the source in the center—the Sun—and themeans with the only force at a distance he knew—magnetism. Unknown at thetime, and not directly measured for three centuries after its first invocation byKepler, this presumptive solar magnetism passes through the planets directlyfrom the Sun and acts according to their mass, their reaction or resistance. Usingthe Mertonian notation that V ∼ F/R, for the velocity to vary inversely with thedistance requires a force that does the same.

Kepler took a much more far-reaching step with his astronomy than simplyfinding regularities in the motions of the planets. In extending the Tychonianprogram of a universe without the spheres and discovering the correct form of theorbits, he felt compelled to apply the new understanding of long range actions offorces in nature—particularly magnetism—to create a new celestial mechanics. Itisn’t important that the specific explanation proved unsuccessful for many reasons,or even that he couldn’t completely break from the earlier dynamical principles.Kepler looms very large because he did something that no one had before: he tooka new cosmological system and attached a physical cause to each part in a way thatsought a universal explanation for cosmic phenomena concordant with everydayexperience.

GALILEO

The Oxford Calculators and their Parisian contemporaries had debated theirphysics in isolation from experience. There were some exceptions, some natu-ral phenomena (meteorological and geological, for example), and in optics andastronomy where it was almost unavoidable. But despite the polemics of Occam,Grosseteste, and Roger Bacon, any discussion of motion and forces was usuallyframed as logical problems and abstractions. But during the fifteenth and sixteenthcenturies, spurred in part by the new exigencies of ballistics and machines and a


Figure 3.3: Galileo Image copyright History ofScience Collections, University of OklahomaLibraries.

growing influence of humanistic formulations of na-ture, experience and experiment gradually movedto center stage.

Accelerated motion is the fundamental concepton which all dynamics was subsequently built.We’ve seen that this was a question that occupiedthe later Middle Ages. But by the seventeenth cen-tury the increasingly numerous contradictions withexperience were becoming more anomalous and lesseasily avoided. The Aristotelian method of requiringcauses fit a dialectic approach. The new approachwas willing to look at the systematics without requir-ing consistency with a specific doctrinal principles.You’ll recall that the cessation of motion of terres-trial things wasn’t really a problem for the Mertoni-ans or their predecessors. In this imperfect world,there are always things resisting the motion so itwould seem a simple thing to imagine that any ob-ject requires a deliberate act to keep it moving. Fora cart, or a river, or a bird this seems reasonable.But there was another, much more serious problem

that this picture simply can’t explain: motion when you aren’t in direct contactwith the object and this was more evident than ever in the ballistics of the earlyRenaissance. There are regularities in projectile motion that seem to be the sameas freefall.

Enter Galileo Galilei (1564–1642), born at the peak of the Tuscan renaissance.His education at the University of Pisa began in medicine in a faculty alreadydeeply committed to the scientific reform of anatomy and physiology; Vesaliushad delivered lectures there and the university had founded the first botanicalgarden for the study of pharmaceuticals. But there was little interest in otherareas of natural philosophy and mathematics, to which Galileo was drawn, and hewithdrew from the formal study before completing his training. Instead, he pursuedprivate study in his preferred subjects, achieving sufficient mastery to obtain aprofessorship in geometry at Pisa in 1587. He stayed there for four litigious years,years before obtaining the chair in mathematics at Padova, which was then partof the Venetian Republic. He remained for nearly two decades, taking privatestudents, lecturing, and advising the state on various technical matters. Manyfeatures of his career resemble his fellow Tuscan Leonardo, then known only asan artist and dabbler but whose investigations in physical science were otherwiseunknown. The comparison is apt. Galileo studied many of the same problems andhad the same broad range of interests. But he was a very public figure in hiswork, a polemicist of talent and erudition, and above all systematic and a superbmathematician. His empirical studies completed the Renaissance project thatbegan with Gilbert and Tycho and the mathematical development that began withKepler without any whiff of mysticism. This involvement as a consultant, duringhis Padova years, parallels Galileo’s development as a natural philosopher withArchimedes. Both had a gift for seeing profound general principles in mundane

The New Physics 49

problems and the concentration needed to develop the ideas, often resulting inpractical mechanisms and proposals.2

Galileo’s Dynamics

During the time in Pisa, and extending through his Padova years, Galileo lecturedon physical natural philosophy. It’s clear from his earliest works—for instance,the unpublished de Motu (“On Motion”) from around 1590—that he knew well thevarious Scholastic force laws proposed in the previous centuries and, with them,the Mertonian rule for calculating accelerated change. But unlike the Scholastics,he concentrated uniquely on the motion and equilibrium of physical systems.Jordanus’ concept of positional weight had been applied only to statics and thenonly using logical arguments. Galileo, instead, conducted experiments with in-clined planes, balances, and moving bodies to discover laws and test conjec-tures.

Why didn’t anyone else try this? Even Leonardo, that dispersive inquisitor,never systematically explored the dynamics of gravitational motion. And nobodyhad realized that the motion is independent of the mass of the sliding body. Abody starting from rest will reach again the same height. This has a remarkablestatic analogy, the hydrostatic balance and the principle that “water seeks its ownlevel.” But even without the concept of work used in medieval statics, there wassomething “conserved” here. A body’s speed at the base of the plane depends onlyon the height from which the body begins. The time to reach any point along theplane also depends only on the initial height, not the inclination of the plane. Twobodies rolling at constant speed reaching the edge of a table will have differentranges depending on their horizontal motion but will hit the ground at the sameinstant. Thus, he reasoned, one and only one of the motions is accelerated, thatalong the vertical. The other continues as it had before the fall began. And further,the accelerated motion is produced by a force acting on a body independent of itsmass. We see the serendipity of working with weights since that depends only onthe acceleration of gravity and the mass.

So for Galileo, freefall is accelerated motion. The magnitude of the speed, notjust the distance covered, changes in time. This is a comparison between distinctquantities. The speed increases directly proportional to the interval of time. Thiswas the odd-number rule, that the proportional distances, sn, covered are in ratiosof odd lumbers for successive equal intervals of time (e.g., in intervals of 1,3,5,etc. for successive equal time intervals), sn+1 : sn = (2n + 1) : (2n − 1), for thenth interval. The weight of the body, and by inference its shape, don’t affect theprocess. Again, this reduces the problem to proportions and geometry. The speedduring an interval of time, T changes from, say, zero to some V . Therefore theaverage speed in this interval increases as T /2. Now since the distance coveredin an interval T is given by the average speed, we find two results: that theproportion of speeds in two intervals is V2/V2 = T2/T1, and that the distancestraversed if the acceleration is constant is L2/L1 = (T2/T1)2 that is the compoundratio L2/L1 = (V2/V1)(T2/T1).3 It is then a relatively short step to the final formof the law, that L = 1

2 gT 2 for a constant acceleration of gravity g. Notice thisactually requires including the dimensions of acceleration, (length)/(interval of


Figure 3.4: An illustration from Galileo’s Two New Sciencesshowing the scaling of a bone as an illustration of theapplication of statics and breaking tension to the size ofanimals. Image copyright History of Science Collections,University of Oklahoma Libraries.

time)2, analogous to the difform difformof the Mertonians. Most important isthat the law is not a law for force nordoes it help yet in defining forces. It’spurely kinematic with no pointer to theorigin of g.

In his youth, sometime while in Pisabefore 1590, Galileo realized that theoscillation period of a pendulum is inde-pendent of both the mass of the bob andits initial displacement from the verti-cal. All that matters is the length of therod or chord on which the mass is sus-pended. This isochronism made a pow-erful impression on him and we know itoccupied his attention for decades. Theobservation that regardless of the weightthe motion is identical for equally longsuspensions was paradoxical within thereceived physical laws. In apparentcontrast to freefall, and more similarto the inclined plane, pendular motionis constrained. The body moves alongan arc whose length is fixed. Since theonly driving motion is its weight, itsacceleration should have depended onthe weight of the bob. It didn’t. Foranyone whose learning had been basedon Scholastic principles this was com-pletely counter-intuitive in the sameway we encountered for a freely fallingbody. But there are two similarities with

the inclined plane. First, once released the body moves through the minimum andreturns to rest the same height from which it started. Its speed increases steadilythrough the lowest point and then decreases as it climbs. If given an initial im-pulse, the body will rise farther, precisely as for an inclined plane, and whenon the horizontal (which for the pendulum is instantaneous) the speed will beconstant. One can imagine that the pendulum breaks at this point, a question thatisn’t in Galileo’s treatises. The period of the pendulum therefore depends only onthe length of the constraint, independent of the initial displacement. This is thehardest point to see without the notion of inertia.4

Galileo’s Statics

Throughout antiquity, and extending into the Middle Ages, there were two, oftennon-intersecting, discussions of force. One was dynamical, the problem of motion.Because after Aristotle the cause of movement was connected with theological

The New Physics 51

ideas, it was the fodder for a vast number of commentaries and doctrinal disputes.The other was more subtle, statics. Although this was certainly the more quanti-tatively mature even before the late antique period, it concerned equilibrium andmore mundane—indeed exceedingly practical—problems, it was far less clearlyappreciated as a general application of the idea of forces. Precisely because itdealt with balance it was thought, at first, to be a separate problem entirely. Themoment when these two strands merged can be pinpointed precisely, the publica-tion of Galileo’s dialog Discourses and Mathematical Demonstrations ConcerningTwo New Sciences in 1638. The two sciences of the title were statics and dynamics.

Recall that Archimedes dealt, in broad terms, with two areas: hydrostatics andsimple machines. In these he was able to ignore any precise definition of forcewhile at the same time making use of weight but he ignored the structure of themachines. How did Galileo consider the problem? First, he began with a storyabout the breaking of a large marble pillar laid on two supporting columns that wasbroken when a third was incautiously added in the middle. This introduced a longexamination of the strength of materials and, in the second day of the dialog, theapplication of static load, torque, and scaling to the general treatment of strengthof materials and static equilibrium. For the first time, a general force, not justweight, was used to describe the breaking of a body.

Strain is the distension, or geometric distortion, of a body. If it’s only onedimensional, a string for instance, this involves only in its contraction or length-ening. If, on the other hand, the object has a finite extension, a beam for example,this involves a distortion in at least two dimensions and requires the concept ofthe lever. It’s odd that even though Galileo discussed the bending moment of abeam, he never proposed any general law for how the form of the body affects itsreaction, nor did he realize the principle that Newton would about 50 years laterwould call the third law of motion, that a body reacts with the same force that anexternal agent exerts. Neither did he realize that if a body is distorted, it mustalso be sheared and this will cause a change in his estimate of the point of failure.Although resistance is defined as the ability of an object to withstand a load—aswe will see later, actually a statement of Newton’s third law of motion: a forceis mutual, that which is acted on resists the force—it isn’t extended to comparemagnitudes.

Let’s examine this example a bit more closely. Galileo analyzed the breaking ofa loaded column by imagining a weight attached at its bottom and hung vertically.This produces a symmetric lengthening of the column, although it might not bevisible, and ultimately will beak it if the weight is too strong. So the stress on thecolumn is uniformly distributed over the area. If there are more attachment pointsfor the material of the column, in other words if the cross section is larger, then thecolumn will have a greater resistance to breaking. OK, so far. Now extend this tothe idea of a beam stuck in a wall. In this case, assume the beam is bending underits own weight. Using the idea that the wall–beam combination acts like a heavylever, in which the weight of the lever isn’t negligible, the point of attachment to thewall below the beam is the fulcrum. In this case, the stress is concentrated there.The resisting force along the cross section of the beam is uniform and the momentarm is now the half-thickness of the beam. The experience from the stretchedcolumn then says the stress—resistance force—is uniformly distributed across


Figure 3.5: The discovery of the presence and motion ofthe satellites of Jupiter from Galileo’s Sidereus Nuncius.This was the first demonstration of bound orbital motionaround another planet and showed the reasonableness ofthe Copernican construction for a heliocentric planetarysystem. Image copyright History of Science Collections,University of Oklahoma Libraries.

the cross section. Although the resultGalileo derived is quantitatively wrongin absolute units, the basic result isqualitatively correct if stated as a pro-portion: the resistance to breaking ofa beam of any length depends on thewidth of the beam in the direction par-allel to that of the load. Thus for twobeams of widths a1 and a2, the resis-tance scales as R2/R1 = a2/a1. It’s aneasy demonstration, take a flexible me-ter stick and allowing enough of thebeam to extend over the edge of a table,first place the broad side on the tableand then perpendicular. You’ll easilysee the difference in the deflection ofthe end. This is the principle that al-lowed for the next step of the Two NewSciences where we see one of the mostnovel applications of physics of Re-naissance natural science. Galileo tack-led a knotty problem, one that wouldsurely have delighted Aristotle himselfand that eventually inspired the Victo-rians (especially after Darwin had in-troduced the concept of evolution bynatural selection): what is the optimaldesign for an animal given its size andmode of locomotion? Galileo didn’t stateit quite this way, but you can see wherehe is going. Because the material itselfhas weight, there is a breaking stressthat depends on the area and volumeof the body, in this case imagine justone bone. A simple uniform geometricscaling, where the dimensions are eachidentically multiplied, m will increase

the mass by a factor m3 while the area will only increase as m2. Thus, without somechange in the conformation of the body—its shape—the animal will be crushedunder its own weight before it can take a single step.5 With this beautifully simplededuction Galileo founded theoretical biomechanics.6

Celestial Phenomena and Terrestrial Physics

In 1609, Galileo was mainly engaged in a wide range of mechanical studieswhile also lecturing on natural philosophy at Padova and taking private stu-dents. Much of his work to that time had been circulated but unpublished,

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although he had published descriptions of a number of his mechanical in-ventions. It was the year news reached Venice of the Dutch invention ofa spyglass, an optical tube that magnified distant objects. It isn’t impor-tant that Galileo did not invent the telescope. He invented observationalcelestial physics. The “cannocchiale” became an instrument of discovery andhe made extraordinary use of it. The results were announced in the SidereusNuncius (the “Starry Messenger”) in 1610. He observed the surface of the Moonand found it to resemble the Earth, showing mountain ranges and valleys, darkpatches that resembled seas and bright spots that he explained as individual peakscaught in sunlight. These discoveries challenged the fundamental principles ofAristotelian physics. As long as the Moon, the closest resolved celestial body,could be considered somehow different from terrestrial stuff it was alright for itto circle the immovable Earth at the center of the world. After Galileo this wasn’tpermissible. The Moon is like an Earth. It must be heavy and must partake ofthe same natural movement as all heavy things. There was no question, though,that it orbits the Earth and after Tycho there were no longer spheres to impede itsdescent.

This contradiction couldn’t be ignored by his contemporaries. The theme of thedynamical connections in the Earth–Moon system would continue to be a majorissue for physics. It is important that the parallactic distance of the Moon from theEarth was already well known that Galileo could put physical dimensions on whathe was seeing on the lunar surface. His geometric construction of how to measurethe height of a lunar mountain was the first example of what we would call “remotesensing.” But more important still was that now the scale of the body could beunderstood, its physical size relative to the Earth was already appreciated but nothow much stuff it contained. Lacking, however, any further explanation for thedynamics, any more than Copernicus himself, Galileo remained content to explorethe extraterrestrial landscape and describe its structure without entering into thedynamics. At least not yet.

In a further extension of the concept of space, Galileo’s resolution of the nebu-lous patches in the Milky Way into an uncountable number of stars gave the firstsignal of a depth to the stellar realm. Increasing aperture and magnification simplyrevealed more, fainter stars in ever direction. Further, they were clumped, not uni-form and continuous as they had appeared to the unaided eye. If the Copernicanpicture were right, he realized, these observations implied there is no limit to theextent of the stellar population, that each is a Sun in its own right (as GiordanoBruno had held).

But the most important result for our history was his chance observation of fourfaint (he called them “small”) stars accompanying Jupiter. With an eye towardfuture employment at the Tuscan court, he named these the Medicean Stars. Theynot only moved with the planet, they circled it periodically. For the first timesince the beginning of astronomy, a motion was observed that could not simply betransformed away by a change of coordinate. The motion was not merely projectedagainst the stars, it was a motion in space around a center that is not the Earth.Galileo completely understood the revolutionary nature of his discovery. Whilethe Copernican system implied a new physics, his observations required it. Againthere was a challenge to the inherited dynamics.


There followed a steady stream of new telescopic discoveries. In 1612, in hisLetters on Sunspots, Galileo announced the observation of irregular, ephemeral darkmarkings—sunspots—that were sited on the surface of the Sun and with which hedetermined the Sun’s rotational axis and period of rotation. This furthered fueledthe challenge to the ancient mechanics: now even the Sun, although luminous,needed to be included in the scheme of a terrestrial physics. The solar rotationalso showed that a body could possess more than one motion proper to itself, afurther argument in favor of the rotation of the Earth. In the same book, he alsoannounce that Venus shows phases like those of the Moon. Understanding that thelunar phases are due to phase dependent solar illumination around an orbit, heshowed that the variation in the angular diameter of the planet and its changingillumination demonstrates its orbit is centered on the Sun, not the Earth. Alongwith the Jovian satellites, there was now no question regarding the heliocentricsystem: it was correct.7

The fruit of Galileo’s observations and thinking about their implications waspublished in 1632, The Dialog Concerning the Two Chief World Systems. Alongwith its orbital motion, Copernicus asserted that the daily motion of the Sun,planets, and stars together require the rotation of the Earth. The seasonal variationof the solar track he explained by inclining the rotation axis to the orbital pole. Afurther consequence of the motion of the Earth was its multiple motions. It bothmoves in its orbit and rotates daily. Neither seems to have a physical consequence.That is, in Scholastic arguments, with the exception of Jean Buridan and Nicholasof Cusa (1401–1464), the motion of the frame is distinct from the bodies withinit. A falling body, in this dynamics, is independent when in motion and if theframe—in this case, the Earth—were to shift by rotation the body would fallbehind the direction of motion. And Ptolemy had persuasively argued that the airwould not move with the Earth and therefore would always produce a wind issuingfrom the direction of rotation. Since these consequences were both obviously incontradiction to everyday experience, common sense implied the stationarity ofthe Earth in all senses. Galileo argued, instead, that motion is shared by all bodieswith the frame in which they are comoving and that no measurement made withinthat frame can dynamically sense its motion. The simplest example is to imagineyourself in an airplane moving at constant speed. If you drop a cup, it falls at yourfeet even if the plane is moving at hundreds of meters per second. This is thefirst example of an invariance principle in physics and is still called “Galileaninvariance.” With it motions in a moving system can be separated into those properto the frame itself and those within the system. Even if, as in freefall, the bodyaccelerates the rotation of the Earth is slow enough that the motion is vertical fora small enough distance from the surface, such as the example.

Despite these discoveries, Galileo didn’t concern himself with an explanationof the heliocentric motion of the planets and the motion of the Jovian moons. Hedid attempt to find an Archimedian sort of explanation for the tides which camedirectly from his advocacy of the Copernican system applying a sort of inertialprinciple. Since the motion of the Earth–Moon system was not uniform relativeto the rotation of the Earth the oceans would execute a daily sloshing motion dueto a torque. This couldn’t succeed as an explanation of the phenomenon but it isimportant that Galileo attempted to apply the principles of force and inertia to acosmic-scale problem. In this as in everything else he did Galileo’s approach was

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distinctly experimental. He was able to use inertia to demolish arguments againstthe motion of the Earth and apply it equally to terrestrial and cosmic motions. Hemade mechanics an empirical physical science. But on the bigger scheme he hadless to say and was not able to appreciate Kepler’s approach to physical astronomyand the question of the origin of the force that maintained the cosmos remainedan open question.

In summary, the result of several centuries of Scholastic discourse was to es-tablish the equivalence of change with the action of a force. Additional effects,additional forces, could introduce changes in the changes, the difference between“uniform difform” and “difform difform.” Regardless of what it is, for the Scholas-tics, as it was for Aristotle and the ancients, the cause of any change requiredsome continual action to maintain constant rate of change. For them, nothing wasconserved in motion. A natural motion is in the essence of the thing but resultsfrom displacement. It is the only form of movement that requires no driver otherthan a final cause within the thing itself. Once the force overcomes some threshold,unspecified but necessary, the motion commences and continues only as long asthe superior driver is maintained. Galileo’s conception was fundamentally differ-ent: the motion, once started, continues, requiring nothing to maintain it in theabsence of resistance. This last requirement is the same as saying: if there is noimpeding action, or force, a body maintains its constant state of motion. This isinertia. The question of weight, or mass, doesn’t enter here because a masslessbody simply doesn’t exist. Instead, extension does make a difference. The state ofmotion is different for sliding or rolling motion, for example, but both have theirinertia and will conserve that state similarly. The same is then true for the celestialmotions, these do not require a separate explanation from terrestrial motions. Norcan they require anything different, since having telescopically discovered thesimilarities with earthly things these bodies should behave similarly.

Galileo’s other great contribution was to introduce quantitative as well as qual-itative elements into the study of nature. When treating the rate of change,the Scholastics could arbitrarily create numbers. Galileo, instead, made actualmeasurements and had real data against which to test his ideas. The way wasnow open for considering a new approach to mechanics in particular and naturalphilosophy in general. If the heavens and Earth are really a continuous whole, thenany experience here applies to the universe as a whole. More importantly, Galileohad set the research program for the centuries to come: the universalization of thephysical laws. Priority disputes aside, what matters isn’t that the effects were new,at times others observed the phenomena before or simultaneously. The differencelies in Galileo’s use of the empirical results, his singular ability to extract generalphysical principles from the experiments and observations and to see their broad-est application in natural philosophy. Before him, the universe had been divided.After Galileo, it would never be again.

Pressure, Temperature, the Atmosphere, and the Forceof the Void

Galileo’s example inspired a completely new approach and his followers wastedlittle time in extending mechanics to practical problems. Castelli and Borelliapplied force concepts to moving fluids, especially canals and the control of


rivers (a perpetual problem in European cities, especially in Italy, during theRenaissance). Galileo’s studies of strength of materials led to works on theoreticalarchitecture and the design of machines and ballistics. I’ll mention here justone of the areas that grew out of this new understanding of forces, the study ofatmospheric phenomena and the behavior of different states of matter.

The expansion of heated bodies was one of the discoveries that led to Galileo’sinvention of a practical device, the thermometer, that consisted of a glass bulkcontaining liquid into which an inverted tube capped by another bulb was placed.When the fluid was heated it expanded up the tube. Its design inspired EvangelisteTorricelli (1608–1647), one of Galileo’s last assistants and pupils, to use a similardevice to study the atmosphere. He replaced the close system with an open pan ofmercury, to increase the inertia of the fluid, and found that the column reached afixed level. He reasoned that the weight of the atmosphere above the pan producedthe pressure that supported the fluid against the vacuum at the top of the column.This is a barometer, the first practical device for measuring gas pressure. He an-nounced the invention to a correspondent at Rome in 1644. This, by extension, alsoconnected with a problem that had bothered Galileo in understanding the causeof the mechanical behavior of materials, the role of “nothingness.” Emptiness wasimpossible for Aristotelian physics. Without resistance a force would produce aninfinite velocity that, obviously, was impossible; Aristotle had removed the voidby a reductio ad absurdum argument. But the atomists had not only accepted theidea but required it. In the first day of the Two New Sciences, Galileo began hisdiscussion of the strength of beams with:

I shall first speak of the void showing, by clear experiences, the nature and extentof its force. To begin with, we may see whenever we wish that two slabs of marble,metal, or glass exquisitely smoothed, cleaned, and polished and placed one on theother, move effortlessly by sliding, a sure argument that nothing gluey joins them.But if we want to separate them parallel, we meet with resistance, the upper slab inbeing raise draws the other with it and holds it permanently even if it is large andheavy, This clearly shows Nature’s horror at being forced to allow, even for a brieftime, the void space that must exist between the slabs before the running togetherof parts of the ambient air shall occupy and fill that space.

He then explained how to measure this force, using a cylinder in which thereis placed a weighted piston using a bucket that can be filled with sand. Air isevacuated from the interior by filling it with water and then, after the air is out,inverting the sealed tube and loading the piston. The force counteracting the loadis then the “force” of the vacuum. This is an ingenious argument. Galileo uses it toexplain the breaking strength of a beam of glass or stone, that this force extendedover the area of the interaction produces a stress that retains the integrity ofthe material. For a continuous medium with weight, the atmosphere for example,this made sense because a reduction of the density in one place produced anacceleration toward the deficit and with such arguments it was possible to produceand explain the barometer.

The theatrically brilliant experiment of Otto von Guericke (1602–1686) at thetown of Magdeburg illustrated this point, again with the inversion of the meaning ofthe void. He evacuated a large sphere, about one meter in diameter, split into two

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hemispheres and tied to two opposing teams of horses. The horses failed to separatethem. It’s important to note that this was, for him, a demonstration of “nothing-ness.” Returning to Aristotle, this showed that space (because clearly the spheresoccupied space) was not corporeal but merely dimensioned. For Von Guericke,this had great theological significance, as he wrote in his Nova Experimentum orNew Magdeburg Experiments (so called) on Void Space (1672):

the universal vessel or container of all things, which must not be conceived accordingto quantity, or length, width, or depth, nor is it to be considered with respect to anysubstance . . . But it is to be considered only in so far as it is infinite and the containerof all things, in which all things exist, live, and are moved and which supports novariation, alteration, or mutation.

This was almost a quarter century after Galileo and now the concept of “forceof the void” had started to become identified with “space.” The force involved amedium between between things, whether they were within and out of air, not inor out of vacuum.8

Pierre Gassendi (1592–1655) continued this line of thought in an argumentthat influenced many of the subsequent discussions of the existence of the void:

Space and time must be considered real things, or actual entities, for although theyare not the same sort of things as substance or accident are commonly considered,they actually exist and do not depend on the mind like a chimera since space enduressteadfastly and time flows on whether the mind thinks of them or not.

If we say that things exist in space and time, both must be separate from matter.Therefore motion exists in both and is a property of the matter within the geometryof the world. We need only a concept of time to complete this, but it is independentof the substance. The void is just that, empty space imbedded in time, and thereis no force to the vacuum. An atomist, Gassendi’s view of material was inheritedfrom the Greek and Roman writers, notably Democritus, Epicurus, and Lucretius,but with significant modifications:

The idea that atoms are eternal and uncreated is to be rejected and also the ideathat they are infinite in number and occur in any sort of shape; once this is done,it will be admitted that atoms are the primary form of matter, which God createdfinite from the beginning, which he formed into the visible world, which finally heordained and permitted to undergo transformations out of which, in short, all thebodies which exist in the universe are composed.

The final phase of this exploration of “nothingness” included the air pump demon-strations of Robert Boyle (1627–1691) and the studies of atmospheric pressureand fluids by Blaise Pascal (1623–1662). Pascal was the first to attempt a mea-surement of the variation of the weight of the atmosphere with height, soon afterthe announcement of Torricelli’s invention of the barometer reached France. In1647–1648, he used a barometer to measure the change in the pressure going fromthe base to the summit of the Puy-du-Dome, near his home in Clermont-Ferand


in central France. He also studied hydrostatic pressure, showing that the effect ofimmersion produced a reaction that is always normal to the surface of the bodyregardless of its shape, a fundamental extension of Archimedian principles, theresults appearing posthumously. By the middle of the seventeenth century theapplication of mechanical laws to the dynamics of both particles and continuousmedia had been started.

CARTESIAN MECHANICS

With Rene Descartes (1596–1650), we encounter a philosopher of Aristotelianproportions whose system encompassed the entire universe and all of its di-verse phenomena. Unlike Leonardo, for whom nature was the ultimate referent,Descartes’ prospectus centers on the observer and the basis of perception. In hishands, geometry became a new, potent analytical tool. Descartes’ introduction ofan algebraicized geometry, quantifying and extending the techniques pioneered bythe Calculators and their contemporaries, reduced Euclid to operations on symbolsand provided a simpler means for producing quantitative results from such argu-ments. Descartes set out the full scope of his mechanical vision in le Monde (TheWorld) and, later, in the Principles of Natural Philosophy. Its range is enormous,from perception and reason to motion and ultimately to the system of the world.It would soon serve as both a foil and model for Newton in his deliberately morefocused work of the same name, with a vastness that inspired a generation to adoptthe mechanical philosophy. Even Galileo had not speculated on so vast a range oftopics in his dialogs, preferring instead to distinguish between the physical andmetaphysical. Descartes did not recognize such distinctions.

The central concept of Cartesian mechanics was, as with Galileo, inertia. ButDescartes asserted that God, not the material thing, is the first cause of movementwho always preserves an equal amount of movement in the world. This is verydifferent from Galileo for whom inertia is an intrinsic property of matter once setinto motion, not a global property parsed out to the different parts of creation.Descartes arrived at this notion by experimenting with collisions, both real andthought experiments. Imagine two balls of equal weight, one at rest and the othermoving. In a collision, the mover comes to rest and the other picks up its motion.If the weights are unequal, there is a redistribution of the impulse between themwith the concomitant division of the quantity of motion. Thus a state of motion canalso produce one of rest and a collision between unequal bodies can even reversethe direction of motion for one of them. This is still not a force principle, as such,because impact can be approximated as instantaneous. From these observationshe deduced three laws of motion. The first states that “each thing as far as in itlies, continues in the same state; and that which is once moved always continuesso to move.” This is the assertion of inertia but with the states of rest and motiondistinguished. Note the wording carefully, when we discuss Newton you will seehow different two versions can be of the same law with just a minor change inthe wording. The second law states that “all motion is of itself in a straight line;and thus things which move in a circle always tend to recede from the center ofthe circle that they describe.” Now we see what Descartes means by force. It iscentrifugal, seeking always to move from the center of curvature. But something

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Figure 3.6: Rene Descartes. Image copyrightHistory of Science Collections, University ofOklahoma Libraries.

must act to maintain the curvilinear motion; thatis the force. The third, and final, law states that “abody that comes in contact with another strongerthan itself loses nothing of its movement; if it meetsone less strong, it loses as much as it passes overto that body.” This came from thinking about colli-sions. He went further in asserting the equivalenceof rest and motion, that these are states of a bodyand that the action required to accelerate a body(change its state) is the same for each. In bothcases, he was describing rectilinear motions, onlychanges in speed. Descartes never asserted thatthese two different states are identical becauserest is relative, there is no concept here of a frameof motion. That was left to Galileo. In a collision,the effect is one of reflection, we would now saythis is in the “limit of infinite mass ratio” that asmall body reflects, without apparently moving thelarger mass. For Descartes a collision was, then,an all-or-nothing affair.

For two dimensional motion, Descartes concen-trated on curves. He stated that such motion requires constraints without using theAristotelian categories (natural, accidental, or violent) and then passes to what hecalls the Second Law of Nature, that “ a moving body tends to continue its motionin a straight line.” As illustration, he uses a stone held in a sling, arguing thatfrom the distance traversed along a plane tangent to the curve that the motion isaccelerated and therefore requires an action, which is provided by the sling andthe hand. The sling and hand (material and final causes, if we were to employHellenistic concepts) are the movement and mover. That when released the ballcontinues in a straight line is because the constraints instantaneously vanish,analogous reasoning then can be transferred from collisions to allow ignoring themoment of escape. Later in the work, he presents it without using the term mass,that a body meeting a stronger one than itself loses nothing of its motion while ifit encounters a weaker one, it gives up its motion. The problem is stated simply:how do we find the force required to maintain a body in motion along a curvedpath? In particular, when the path is a circle? Descartes realized the motion ofa body is accelerated in some way when it moves in a circle, but he dismissed achange in direction as something that requires a constraining force. Think of thattethered ball. If we ask what the speed is, then along an arc of a circle—providedthe length of the string remains unchanged—the time to cover equal portions ofthe arc is the same (constant circular speed). Yet there is a tension on the string.That comes from resisting the tendency of the ball to escape by moving away fromthe center.

Although he began as a Cartesian enthusiast—indeed he was Descartes’student—Christian Huygens (1629–1695) provided, through the repudiation ofsome basic Cartesian principles, the first quantitative law for force in circularmotion. In 1657, Huygens dealt with this by requiring the motion to be an outward


Figure 3.7: Figure from Descartes’ Principia showing hisexplanation for centrifugal force. A body tends to recedefrom the center of a curve, the sling maintains the motionaround the curve against this. While this is true in themoving frame, Newton demonstrated that the unconstrainedmotion is really along the instantaneous tangent to the curvein a straight line, not radially outward. Image copyrightHistory of Science Collections, University of OklahomaLibraries.

tendency of the body in contrast to thetension of the sling. Thus was formedthe concept of centrifugal force, whichwould become a central concern forNewton. Huygens took the change indirection, not simply the speed, to bean attribute of accelerated motion and,since this is outward, took the acceler-ation to be similarly directed. The rateat which a body recedes from the curvealong the tangent plane clearly accel-erates, the projected velocity varyingin magnitude to reflect the constrainednature of the motion, the conjunctus.The acceleration depends on the tan-gential speed v and the radius of thearc, r , varying as v2/r . You’ll noticeI’ve called this an “acceleration,” an-ticipating later developments; the ten-dency is a vague thing since the defi-nition of the mass is left unresolved inDescartes and Huygens. But there is amore important feature. This is an out-ward tendency from the point of viewof the observer moving along the arc.The body, now freely moving along thetangent plane, seems to be receding atincreasing rate while, in fact, it is mov-ing at constant speed and in a constantdirection.

In Cartesian celestial mechanics theuniverse is container filled with a com-plex dynamical fluid, an ether, but notthe sort imagined by the Hellenistic

philosophers. This one is not only in constant motion but filled with vortices.It was his explanation for the attraction that binds the planets to the Sun and theMoon to the Earth. It’s justification comes from a familiar phenomenon. Take acup of water and put tea leaves into it. Stir it so the tea moves to the sides of thecontainer. So far, no problem, this is a centrifugal force that Descartes had pos-tulated is an essential property of circulating matter. But if you stop the stirring,the leaves will move to the center and bottom of the cup. The fluid at the walls ismore slowly than in the center of the vortex. It seems to be an attraction that drawsthe leaves toward the center, just as gravity does for heavy bodies. To Descartesthis was a powerful analogy. As Giordano Bruno had before him, Descartes wasseized with the image of each star being a sun at the center of a vast sea of cosmicvortices each supporting possible planetary systems. The inspiration of Galileo’sastronomical discoveries and Kepler’s laws of planetary motion combined with a

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dynamical driver completed the construction. Cometary paths could be irregularbecause they pass from one vortex to another. The Moon is drawn to the Earth byits local vortex while the pair orbit the Sun for similar reasons. Although Descartesmade no attempt to check whether this picture is really consistent with Keplerianmotion and the harmonic law, it was enough to provide a compelling physicalconstruction and a hypothesis for the origin of gravitation without natural motion.

On the other hand, the essentially Aristotelian concept of a force still applies,this is a medium that is actively driving the motions so the inertia of circular motionthat Galileo had introduced is missing. Although not a magnetic driver, the vorticesbear a striking resemblance to Kepler’s dynamical model. Furthermore, the natureof this medium is unexplained, whether it has gravity itself or only produces it. Thisconcept of a space-filling vortical plenum remained an attractive explanation fornearly 150 years. It satisfied almost all explanatory requirements even if it lackedquantitative details. It spanned the gap between Scholastic mechanics and the newphysics. It impeded, however, the acceptance of the existence of a vacuum, sinceif the universe is completely filled only relative augmentations or diminutionsare possible, and in adopting a continuum denied the existence of atoms. Hencethe system explained well the motion of the planets within a heliocentric system,linked cosmology, celestial mechanics, and rather general laws of motion even asit left huge conceptual and foundational holes.

ROBERT HOOKE: THE FORCE LAW FOR SPRINGS

Among his numerous optical, naturalist, linguistic, and astronomical studies,Robert Hooke (1635–1703) made a vital contribution to mechanics with a com-monplace discovery, his elastic law announced in his Cutlerian Lectures publishedin 1679: the extension of a spring, �L, depends linearly on the weight attachedto its end and the material of which the spring is made, weight = constant × �LIf we replace the weight by any other action, while requiring that the spring thatit can only stretch in response to an applied force, we can replace the weight byany force. Then we arrive at Hooke’s statement that strain, the deformation of anelastic body, is directly proportional to stress, the impressed force.

Consider for a moment the number of concepts underlying this very simple re-sult. Before the promulgation of Hooke’s law, the measure of weight was essentiallyalong Archimedian lines with the level or steelyard. Although using unequal arms,the principle is unchanged from the early mechanical treatises. The two forcesare identical since they’re both weights and therefore gravity alone is involved.You don’t need to consider either the material (unless the level or arm distorts)or anything else about the constitution of the material. But if you use a simpleelastic law for the measure of the force, you’re actually contrasting two differentforces—elasticity and weight—and saying that one is the reaction to the other.This is not the same as a beam balance and the difference is very important. Weare actually making a generalized statement, that all forces are equivalent, andthen developing a technique based on this. Notice also that the spring balance usesthe reaction—extension—of the spring and not its acceleration. But if we releasethe end, the spring contracts, or in the language we have already used “changesits state of motion,” by accelerating. This is easily seen. It also doesn’t matter what


form we choose for the spring, helical or linear, or whether the medium is simplyelastic (so we have just one dimensional deformation) or helical (where we twistand extend the spring simultaneously). Elastic deformation actually affords a con-nection with the discussion in Day 1 of the Two New Sciences, when Galileo talksabout the breaking of a beam, because we now have a model to complement hismore qualitative arguments from geometry. In fact, Galileo had already assumedsomething like a spring law for building up the structure of an elastic body (thefibers of the wood act as individual, extendible elements of the material) but withHooke’s law we have a direct, verifiable, quantifiable principle.

NOTES

1. Although contemporaries thought this preface, which presented an even strongerqualification of the work as a hypothesis, was written by Copernicus himself, Kepler latershowed that it was actually a separate, clandestine addition to the work a the time of itspublication.

2. For instance, two of Galileo’s earliest productions were works on the hydrostaticbalance, bilancetta, and a device for computing ballistic range and flight, the militarycompass. He also served as a consultant in practical, engineering matters to the Venetiansenate much as Archimedes had to the ruling class of Syracuse.

3. Stillman Drake, the principal biographer of Galileo of the last century, made an ex-cellent point about the use of algebraic notations in understanding the Galilean argument.Any ratio was held, following Euclid, Elements Book V and Eudoxus, to be a comparisonof physically similar (commensurate) quantities. To belabor the point for a moment, thismeans you can compare distances and times separately but not together because theyare not the same thing. While now we employ the algebraic notion of units, as we haveseen motions are decomposed into space and time separately and these were not on thesame footing in the seventeenth century. Thus, to arrive at the quadratic relation for thetime dependence of displacement, and then to extend this to a general dynamical law foracceleration, requires using a compound ratio of two separate quantities.

4. We consider the velocity of the body at any moment to depend only on the time.The distance covered along an arc, s, is the angle through which it moves, �θ times theradius of the circle, R. Since the weight is both falling and moving horizontally, and thatsecond motion is only due to the constraint, the vertical distance of the fall should dependon the square of the time. We note that, for a small angular displacement, the bob is raisedthrough a vertical distance R�θ . Like a body on an inclined plane, the positional gravitywill be larger at the beginning than at the end of the motion.

5. In the twentieth century, it might have been more interesting if science fictionwriters had read the text of Two New Sciences more carefully. Godzilla is a scaled upversion of a dinosaur. The Amazing Colossal Man and the Fifty Foot Woman are examplesof how little attention was paid to such consequences (a lovely essay from 1928 by J. B. S.Haldane, “On Being the Right Size,” is a notable exception). Yet Galileo himself pointsout the problem when discussing marine creatures, since he shows that on land they will beunable to function, being without the hydrostatic support provided by the water in whichthey live.

6. You may object that Galileo was working almost 150 years after Leonardo da Vincihad made similar proposals. The difference, as I’ve emphasized, is that Leonardo neverprepared anything systematic, or even synthetic, on his physical investigations. He’dthought about publishing them but never got around to it, preferring always to be wanderingin uncharted intellectual territory. His reputation by the close of the sixteenth century was

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as a supremely gifted artist, if not eccentric, inventor and applied mechanician. Many ofGalileo’s derivations and ideas were along the same lines but that is an indication of howconvergence occurs in scientific problems rather than an intellectual indebtedness.

7. It is, however, important to note that this observation is not as convincing asthe Medicean satellites. The Tychonian system, kinematically equivalent to Copernicus’but with a geocentered world, also implicitly predicted the correct phases and diametervariations. This isn’t at all surprising since the construction is purely geometrical. Butthere was a high price with this. Still lacking any dynamics, it further divided the behaviorsof the celestial bodies by forcing two of them to orbit the Earth while the rest, for unknownreasons, revolved around the Sun alone.

8. The translations in this section are from Grant (1981).


4

FROM THE HEAVENS TOTHE EARTH

You sometimes speak of Gravity as essential and inherent to Matter. Pray do notascribe that Notion to me, for the cause of Gravity is what I do not pretend to know,and therefore would take more time to consider it.

—Isaac Newton, 2nd Letter to Richard Bentley, 17 Jan. 1692–3

We now come to the pivotal figure in the history of the concept of force, IsaacNewton (1642–1727). He has been called the last of the magicians by Keynesand, indeed, Newton was a person standing between ages. He was a dedicated—obsessive—alchemist and theologian, a hermeticist, and biblical chronologist. Buthe was also a profound experimentalist, inventor, mathematician, and observer.Although he held a university chair, it was more titulary than faithfully prosecuted.He served as advisor to his government, Master of the Mint, and president of theRoyal Society. He was both a theoretical and practical natural philosopher and theparadigm of both in his time. His role was transformational, no area of physicalscience or mathematics escaped Newton’s interests. And none since Aristotle—not even Copernicus, Kepler, Galileo, or Descartes—so completely explicatedthe mechanisms comprising a single system of the Universe. We will linger inNewton’s company for some time and then examine how his physical heritagepassed to the next generations in the eighteenth and nineteenth centuries.

THE BEGINNINGS: THE KEPLER PROBLEM

We know from his early jottings, the Trinity Notebook and the Wastebook, thatNewton began to think very early about the motion of the planets, even before thePlague Year of 1665–1666. By his own testimony in his later years:

In the same year I began to think of gravity extending to ye orb of the Moon and(having found out how to estimate the force with wch globe revolving within a spherepresses the surface of a sphere) from Kepler’s rule of the periodical times of the


Planets being in sesquialternate proportion to their distances from the centres oftheir Orbs, I deduced that the forces which keep the Planets in their Orbs mustreciprocally as the squares of their distances from the centres about wch theyrevolve: and thereby compared the force requisite to keep the Moon in her Orb withthe force of gravity at the surface of the Earth, and found them answer pretty nearly.All this was in the two plague years of 1665–1666.

Distinct from Kepler, however, Newton realized that the force would act in thedirection of fall, not in the tangential direction. We will later see this in the secondlaw of motion but for now, we concentrate on how he exploited this insight. If theMoon would move, inertially, in a straight line in the absence of the Earth, thenit must be attracted to the center just as any body on the surface is. Ignoring theeffect of the Sun to which the Moon is not bound, we can ask what would be thetime for fall were the Moon to cease its orbital motion (the origin was which isanother problem). The time for fall of any body on the Earth’s surface dependson its height, so knowing the distance to the Moon, about 60 Earth radii, Newtoncalculated the time compared to that needed for a body to reach the center of theplanet. This agreed well enough with his hypothesized force law to be encouraging,but not precisely. There are several possible explanations for why the disagreementpersisted but the most direct are the incorrect distance to the Moon and the wrongradius for the Earth.

For most of the intervening years between 1666 and 1684, Newton occupiedhimself with other matters than mechanics. He was deeply engaged in optics,alchemy, and geometrical analysis. His announcement of the prismatic separation,and reconstruction of white light, the discovery of interference effects (known asNewton’s rings), the invention of the reflecting telescope, and the systematicexamination of optical systems seem far from the preoccupation with celestialdynamics that so deeply engaged him during his student years. But the stage wasbeing set by long, solitary years of extending his experience and meditation.

A chance encounter triggered the revolution. By his own account, EdmundHalley (1656–1742)—with whom Newton was a familiar from the Royal Society—had been studying comets and visited Newton to ask advice on how to treat theirmotions. On asking what form their paths should take, he received the answerthat they should be ellipses. Although, given Kepler’s laws of planetary motionthis doesn’t seem an especially surprising response, Halley reported that he wasastonished and, on probing how Newton knew this received the answer “I’vecalculated it.” Halley requested a demonstration and Newton obliged, first witha small account on motion and then, on the urging of both Halley and the RoyalSociety, embarked on a more comprehensive exposition. It would take him yearsbut, as the historian R. S. Westfall put it, “the idea seized Newton and he waspowerless in its grip.” The result in 1687, after nearly three years of unceasingeffort, was the Principia.

THE PRINCIPIA MATHEMATICAE NATURALIS

The structure of the Principia hides these developmental twists very well andpresents the final layout in logical order.1 Much of the layout of the logic follows

From the Heavens to the Earth 67

from, and supersedes, Descartes’ physical treatises. The logical organization ofthe first chapters of the first book is Euclidean and deductive. Newton opens withthe definition of mass, likely the thorniest conceptual issue in the whole work,defining it in effect as the quantity of stuff of which a body is composed. Extensionisn’t important, as it had been for Descartes. Instead, this is an intrinsic propertyof material. He also explicitly identified a mass’s quantity of motion, what is nowcalled the momentum, with its inertia.

Newton then presents the three fundamental laws of motion. The first states that“Every body perseveres in its state of being at rest or of moving ‘uniformly straightforward’ except insofar as it is impelled to change its state by forces impressed.”This is Newton’s statement of inertia, the motion in the absence of any externalperturbation. Having defined mass as the fundamental quantity, Newton took thestep at the start of creating two separate quantities: the momentum or quantity ofmotion, and the mass. The second law is that “a change in motion is proportionalto the motive force impressed and takes place along the straight line in which theforce is impressed.” This is Newton’s definitive statement of what force means.Two quantities, kinematic and dynamic, are connected. The acceleration producedby any agent is always in the direction of the imposed force. For the first time,this law makes clear that the force can be deduced by its effect, that the inverseproblem can also be equally well posed—to find the force if the change of stateof motion is known. It also introduces the definition of force as a rate of changeof the state of motion in time. Finally comes the third law, that “For every actionthere is always an opposite and equal reaction; in other words, the actions of twobodies upon each other are always equal and always opposite in direction.” Thisis why Newton required some definition of mass because it enters implicitly in thethird law. If there is any external agent acting on the system, the reaction of thebody depends on the mass—or inertia—and is a phenomenon independent ofthe composition of the body or the nature of the force. It is always directed inreaction to the force and its acceleration is proportional to the mass. Actually, thethird law restates the second in a far more general form, asserting the universalityof the reaction.

A series of corollaries follows in which Newton deals with the composition offorces in analogy to that of the motions. Since the motions are decomposable intospatial components, those which change display the direction of the force andtherefore, a body acted on by two forces acting jointly describes the diagonal of aparallelogram in the same time in which it would describe the sides if the forces wereacting separately. Colin Maclaurin (1698–1746) in the first popular exposition ofthe Principia for an English speaking audience, Account of Sir Isaac Newton’sPhilosophical Discoveries (1748), expanded this to include the explicit case ofdecomposition of velocities, which Newton had assumed was already familiar tohis readers.

To extend this to the case a of a self-interacting, one in which all the individualmasses act mutually on each other, Newton required a modified version of thefirst corollary, “the quantity of motion, which is determined by adding the motionsmade in one direction and subtracting the motions made in the opposite direction,is not changed by the bodies acting on one another.” This was completed by theproof that would eventually be important for constructing the mechanics of the


solar system, that “the common center of gravity of two or more bodies does notchange its state whether of motion or of rest as a result of the bodies acting uponeach other; and therefore the common center of gravity of all bodies acting uponone another(excluding external actions and impediments) either is at rest or movesuniformly straight forward.”

The canvas is already starting to show the big picture. In Bk. I sec. 2 Prop. I.,Newton proposes “to find centripetal force.” This proposition, the centerpiece ofthe entire study of gravitation, derives the inverse square for a general motion basedonly on the conjunctus and a continuous curve. But it masks a subtle argumentabout the continuity of motion itself, not just the geometry of the trajectory,and already introduces the differential calculus that was necessary to connectaccelerated motion with its geometric result. In Prop. 5, he seeks to show how,“given, in any place, the velocity with which a body describes a given curve whenacted on by forces tending toward a common center, to find that center.” Finallycomes the moment of triumph, the answer to the question that Halley posed twoyears before. In Prop. 10, Problem 5, Newton seeks to show that a closed conicsection, an ellipse or a circle, is the unique trajectory for a closed motion abouta central gravitational source. “Let a body revolve in an ellipse; it is requiredto find the law of the centripetal force tending toward the center of the ellipse”and in Prop. 11: “Let a body revolve in an ellipse; it is required to find the lawof the centripetal force tending toward a focus of the ellipse.” With Prop. 17 wefind the final statement of the inverse square law and in Prop. 20, Newton solvesthe general problem of trajectories. This is extended in the next proposition tothe hyperbola, and then to the parabola. His answer is the consistency proof thatonly with an inverse square law can Kepler’s orbits be recovered. Then, in thescholium, Newton added that the conic sections are general solutions includingthe parabola and explicitly citing Galileo’s theorem.2

Later propositions deal with the determination of orbits from finite numbersof points, a problem later dealt with by Gauss for the determination of orbitsfrom observations. To display the logical structure of the argument, it sufficesto outline the rest of the book. The exposition is organized as follows. In Prop.30–31:, Newton shows the calculation of the orbits, the Kepler equation; in Prop.32–39: he treats linear descent (that is, freefall); in Prop. 40–42, he generalizesthe curvilinear motion to treat arbitrary central force laws; in Prop. 43–45, heshows how perturbations produce a motion of line of apsides3 of an otherwiseclosed orbit; Prop 46–56 return to the Galileo problem of pendular and oscillatorymotion; Prop. 57–69 expand the application from central forces between bod-ies to systems of particles, orbits determined by mutual gravitational attraction;and finally Newton presents the solution for the gravitational field of spheri-cal (Prop. 70–84) and non-spherical (but still nested ellipsoidal) bodies (Prop.85–93).

Here we end our summary of the first book of the Principia. Although there areseveral additional sections, you see both the structure of the argument and the useof the force concept. Newton had, by this point, assembled the machinery necessaryfor the general construction of the cosmic model of Book III, the System of theWorld. But there were still several points to deal with, especially the refutation ofCartesian physics.


The Banishment of Centrifugal Force

With the realization that motion of the planets can be explained with the uniqueaction of a central force, the need for a centrifugal force disappeared. Yet thisconcept persists in daily life, and it is instructive to see why. Newton adopted asingle frame of reference, the Universe at large, and his ideas of space and timewere imbedded in his methodology. There is an absolute frame relative to whichanything is said to change. This includes velocity as well as acceleration (which,in turn, includes changes in either speed, direction, or both). Thus, for a bucketwhirled around a center and attached with a string, any fluid within the bucketappears to be driven to the bottom because it is forced to move relative to thisuniversal coordinate system. For Newton, this was not mysterious, the change indirection was produced by a centripetal force acting through the tension of thestring and the trajectory of the contained fluid is constrained by the container. Ifthere were to be a hole in the bottom, the fluid would exit by moving tangentiallyto the path of the bucket (you might think here of the spin-dry cycle of a washingmachine). The material has been torqued so it has angular motion, but the torqueneed not be maintained to keep the bucket in “orbit,” nor do the planets needto be continually pushed along their orbits—they only need to be constrainedby a radially directed force to remain within the trajectory around the centralbody.

Now, in contrast, the motion within the bucket is described very differently.In this frame, we are no longer moving inertially, so there appear to be forces.The water doesn’t “seek” the walls of the container, instead the walls get in

distance coveredin a time ∆t

∆V

V

V

R

R

center

Figure 4.1: The Newtonian construction of centripetal force. A particlemoving along a circular path centered at any moment at some distance Rhas a tangential velocity V. The direction of this motion changes althoughthe speed remains constant. The acceleration, the direction of the changein the tangential component, is toward the center. The force is, therefore,in the same direction according to the second law and equal in magnitudeand opposite to the change in the motion by the third.


the way of the water, which is simply continuing to move. The bucket not onlyconfines the motion, it also creates an internal reaction in the fluid that is equal andopposite—in the frame of motion—to that which it would have were a force of equalmagnitude to that produced by the motion of the frame applied from outside. Thefluid is “completely ignorant” of its frame of motion, as is a person confined withinthe same moving bucket, so because there is an acceleration there’s a “force.”4

It’s the laws of motion that permitted the derivation of the universal law Newtonsought. The first step is to ask, if we have circular motion, that is the observedacceleration? Moving around a closed curve of constant radius, the direction ofmotion changes continually. This is just what Huygens had found, but with abasic change of viewpoint: instead of this being the force, it is the accelerationexperienced by the test mass, m. For a constant velocity, the body—Newtonrealized—must be continuously accelerated because it is continually changingdirection although not speed. Thus, for a convex curve, the direction of the changealways points toward the center of the curve. If this is produced by a force, thatforce must be continuously drawing the body inward, hence it is always attractive.With the direction of the change fixed, it follows that the deviation gives a forcethat points toward the central mass and depends on the quantity of motion mv ofthe moving body as:

(Force) = −mv2

r

The second law of motion, stated a bit differently, asserts that if we know theacceleration, the force producing it lies in the same direction and has the samemagnitude per unit mass. But where this comes from is particularly illuminating.Let’s sketch the proof. Imagine a curve with a local center C and take the arc ABto be subtended by an angle AC B and the chord AB. Take the radius to bisect thechord at some point P . Then at the point of intersection with the arc, Q, connectthe chord to form QA and QB. Since the construction is symmetric about the lineC Q we’ll use only the right hand side. Now since the radius is perpendicular tothe bisected chord, QP : QA = QA : QC so we have QP = QA2 : QC. But sinceQC is the radius to the center R and QA = V�t with V being the speed, thenit follows that the deviation of the direction of the motion is the measure of thecentripetal force and the quoted law follows. Although this is formally the sameas Huygens’ result, the direction is opposite: this is a force toward the center C,not a conatus produced by a centrifugal tendency. While the result of this lawhad already been known 20 years before Newton’s work, it was the Principia that,by reversing the direction, forever changed the concept of gravitation as a force.Yes, this seems rather flowery prose but I know no other way to imagine what itmight have felt like to have this insight and you would do well to contemplate itssignificance.

The force must increase with increasing mass of the central body so the acceler-ation it produces is independent of either the mover’s mass or time. Furthermore,the force must act only toward the center of the body. In fact, even if each parcel ofmass attracts the body, only that toward the center is needed. We could imagine abody so distant from the mover that its size is essentially that of a point. Then the


proposition is simple, we need only consider the radial distance. For a sphericalbody of finite size, the portion closer to the mover has a greater attraction but,for a fixed solid angle, there is less of it, while that at greater distance exerts aweaker attraction but there is more of it. In the end, these sum to the force all of themass would produce if all of the mass were concentrated at the precise center of thebody. To extend this, Newton introduced what we could call the interior solution,the one that would be exploited later by Laplace and Poisson: for a homogeneousset of nested spheroids—without lumps so they are symmetric—only the massinterior to any radius attracts the test mass and any external matter produces a netcancellation. Again, this is an effect of summing (integrating) over surfaces of fixedsolid angle. We can step back, somewhat anachronistically, and write that dimen-sionally the acceleration is R/t2 (where here R and t are intervals of space andtime, respectively) so the force must vary as 1/R2 since the velocity is R/t . Putmore directly, since we know from Kepler’s harmonic law that the orbital periodvaries such that R3/t2 is constant and the same for all orbiting bodies we arriveat the same result. This had been the first route Newton had taken, comparing theacceleration of the moon with that of a falling mass on Earth scaled to the lunardistance and found good agreement, at least good enough to pursue the argumentfurther. By this point in the writing of the Principia, twenty years later, he knewthe answer.

Universality

At the heart of the Principia lies a radically new conception of the Universe. Notonly is it governed by physical laws, or natural compulsions, it is all the same.The structure and operation of the world can be understood by a simple, universallaw, gravitation, that not only explains the phenomena but predicts new ones.How different this is from the medieval world. This is not from a hypothesis. Itis from a theoretical generalization of empirical results that now are predicted tobe consequences of the laws. The motion of the planets is the same as terrestrialfalling bodies. The tides have yielded to a single force that is the same oneresponsible for the motion of the Moon. There is no need to posit a mechanismto transfer the force, although it seemed Newton couldn’t restrain himself anymore than others in seeking one out, nor do we need to understand the source ofthis force. The methodology is now completely different, we will hence derive theproperties of matter from the observation of interactions by applying a few lawsand then deriving their effects.

To understand the revolutionary calculation contained in Book I of Principiaconsider how the the laws of motion have been applied. The solution to circularmotion, more than any other proposition in Principia, shows how dramatic isthe effect of Newton’s laws of motion. The change in the fundamental picturecomes from the role of inertia. Were a body to accelerate centrifugally, that is,attempting to recede from the center of motion, its behavior would have to be anintrinsic property of matter. Instead, by fully grasping what inertia means, Newtontruly dispensed with any such occult properties of matter, in the very sense of thecriticism later leveled against him by Leibniz. The body does not seek to moveanywhere other than to preserve—notice, this is not conserve—its state of motion.


With this as the definition of inertia from the first law, with no explanation eitherrequired or given since he affirms it as a law, Newton was able to explain theapparent centrifugal action by a simple change in reference frame. This is not howit is described in the propositions of Book I but that presentation amounts to thesame statement. Instead, taking the acceleration to be the deviation toward thecentral body relative to tangential motion, which is the instantaneous continuationof the inertial motion, then the force becomes radial and acts only between thecenters of the masses.

The striking result is that the answer is identical. Only the sign is changed forthe force. But what is far more important is that this is a general result, not oneconfined only to circular motion: since the mover is deviated at every instant, andthe limit can be taken relative to a continuous curve as Newton showed in Prop. 8,there is no restriction. Any conic section can result depending only on the initialconditions. Since the forces are only between the bodies and neither external tothe pair nor tangential, as he had shown extending the treatment from Galileo andfrom ballistics, the limiting case for a constant acceleration will be a parabola andtherefore any eccentricity can result. Recall that the limiting case for a criticalhorizontal velocity, that the radius of the orbit is the same as the radius of theEarth, is an interesting variant on the Galilean argument. Since for a circular orbitthe distance from the gravitational source is constant, this is actually motion underconstant acceleration and shows that the limit of the parabola or ellipse going tothe circle depends on the tangential speed alone.

Newton’s illustration of the limiting orbit for a cannonball fired horizontallyto reach a limiting speed is a spectacular graphic as well as a precise statementof the problem, one that can easily be extended to an escape condition is theinitial tangential velocity exceeds the orbital limit. This is, however, not enoughto explain the motion of the Moon or any other pair of orbiting celestial bodies, in-cluding the Earth–Sun pair. While the first and second law suffice, without modifi-cation, to yield the solution for an enormous mass ratio between the mover andcentral body, the third law also provides its extension to finite, near unity, mass ra-tio. By saying that the force is mutual and opposite, that the two bodies experienceequal and opposite forces but consequently accelerations in inverse proportion to

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Figure 4.2: Newton’s example for orbital motion from the Principia. A pro-jectile is launched with increasing initial horizontal speeds. Eventuallythere is a critical value at which the body falls at the same rate as it dis-places horizontally, maintaining a constant distance. Thus circular motionis always accelerated—it’s always falling but never hitting the ground.


their masses, Newton made the critical step required for a celestial mechanics. Thecenter of mass, or center of gravity, remains unmoved in the absence of an externalforce. For the Earth–Moon system, this means the two bodies, if stopped in theirorbits, would fall toward a fixed point in space. Notably this seems to endow spaceitself with a special property, one that was later to exercise Newton’s ingenuityand later commentators in much the way the discussion of space and place wascentral to Aristotle’s thinking. The existence of this point is another expressionof inertia, that the moving system has itself a common center that will have itsown trajectory if a third body or external force acts. So for the Earth–Moon–Sunsystem, this becomes the explanation for the continued binding of the Earth andMoon while their common center orbits the Sun. The same is true for the otherplanets and their respective moons and rings.

The presence of this external force, in the Earth–Moon case, also disturbsthe lunar orbit. While various deviations in the orbit had been known from theearly modern period, and even Ptolemy was constrained to add special kinematicdevice—the moving equant—to account for the precession of the apsidal line ofthe lunar orbit, Newton had discovered its dynamical explanation. The relativemagnitudes of the forces of the Earth and Sun on the lunar motion reduced the solarinfluence to a mere perturbation. Were it stronger, the Moon could instead orbit theSun and be perturbed by the Earth as would happen with any of the other satellitesystems. But with this realization, the scaling of the solar mass and, therefore, thedensity of the two principal bodies affecting the Moon could be determined andthis is included in Book 3 of the Principia. We will return to this shortly.

The logic of Newton’s argument is then clearer if we think to give this notionof a center of mass a prime role in his thinking. First find what we would nowcall a reference orbit and then, after demonstrating that it must be one of thetwo closed conic sections, show how it will be disturbed in the presence of anyother gravitating body, i.e., mass. By the end of the first book the reader is reallyprepared for Book 3, the System of the World, which is the applied part of thePrincipia. But before taking such a step, and in particular because the Cartesiansystem of fluid driving in the place of his gravitational force was so widely diffusedin the philosophical community of his time, Newton felt compelled to not onlyexamine the application of the force law to fluids but also to use it to refute thevortex picture. The study is as important to his system as it is to demonstrate theenormous reach and power of the new mechanics.

In Book I, Newton solved the two principal problems of celestial mechanics, atleast to a first approximation. One deals with the motion of the center of mass of atwo body system in the presence of perturbers, the main issue in the stability of theEarth–Moon system in orbit around the Sun but also the central issue regardingthe overall stability of the Solar system. He showed that the centers of mass of thetwo systems can be treated as pseudo-mass points whose motions are determinedby the relative intensities of the forces of the nearby—and distant—masses ofwhatever magnitude. The second, far more subtle problem is related to extendedmasses: what is the gravitational force of an extended body, that is when thedistance between the masses is comparable to their size? He demonstrated, ina series of propositions accompanied by examples, that the attraction on a pointmass is the same as if the total mass of the more extended sphere—the one case


he could resolve directly with a minimum of approximation—is concentrated atits center. To go to progressively more realistic problems, he then showed that thesame is true for two spheres, whatever their relative density distributions, therebypermitting the solution of the Kepler problem even when the two objects are verylarge. Finally, and this is perhaps the most beautiful of the results at the end ofthe first book, Newton showed that the attraction of the mass within a sphere onits superficial layers is as if the mass is concentrated at the center and only themass within a given radius attracts the overlying layers.

This is one of the few examples in Principia where Newton resorted to symmetryarguments. He began with a hollow shell, an idealization that allowed him to usethe inverse square law explicitly to show that the force cancels exactly within thesphere for a mass placed anywhere inside. His argument is simple and elegant.The solid angle subtended by any piece of the shell is constant but the mass withinthat cone depends on the area. Since that varies as the square of the distance, whilethe force produced by the shell varies as the inverse square of the same distance,the two intersecting cones produce equal and exactly oppositely directed forcesand therefore cancel. The same argument served, with the asymmetry of the testmass being situated outside the sphere, in the later propositions that demonstratethat the force is as if all the mass were concentrated precisely at the center ofthe body. Then, taking two differently stratified spheres, he extended this to twofinite bodies. But the tour du force came with Newton’s demonstration that this sameargument also holds for nested homeoids, concentric spheroids with precisely thesame symmetry. Thus for any density distribution it was possible now to derive thegravitational force on any internal shell. With this, he passed from the dynamicalto the static problem and introduced the line of research that opened the structureof the Earth, and of all celestial bodies, to remote sensing. With this construction,it was possible to study the inverse problem although, as we will see, that took onemore conceptual change to complete the picture.

A weakness of Newton’s calculation was realized and corrected rather soonand this was an essential extension to the theory. Imagine a pointlike mass inthe vicinity of an extended body. Then by linear superposition, the fact that wecan simply add the gravitational fields produced by individual masses each in itsappropriate direction, each piece of the body attracts the mass (and vice versa)according to its distance. Thus although the center of mass is precisely definedwith respect to the two bodies, the resulting gravitational field depends differentlyon the between the bodies than a simple inverse square law. This is because someparts of the body are closer than others and also some are farther from the line ofcenters. The resulting representation is a series of polynomials in the two relevantangles, the azimuthal and the meridional, named after their originator Legendre.It was, however, Laplace who saw the generality of their properties and showedthat these are the solutions to the gravitational field for an arbitrary body whenexpressed in spherical coordinates. An orbit, even in the two body problem, isnever a closed simple ellipse. For any body of finite radius, R, there are alwaysterms of order (R/r )n for any arbitrary n and distance r . What Legendre had shownwas that “his” epiphanous polynomials resemble a distribution of point massesacting as if they are dipoles, quadrupoles, and higher order. Thus, for instance, abar is a sort of dipole although with two equal “charges.”


Refutation of Cartesian Methodology

After its concentration on orbital mechanics, it might seem strange that the nexttopic addressed in the Principia is the motion of a continuous, fluid medium andits action on internally moving bodies. Yet you should recall here the audienceto whom the work was addressed. The subject occupies virtually the whole of thesecond book. It was one thing to have treated the behavior of things, but causes werelacking for the forces, especially gravity. While various hypothetical explanationscould be supplied for most material, contact forces, gravity resisted. For one thing,acting apparently at a distance, gravitation required some means for transmission,yet nowhere in the first edition does Newton explicitly say anything about this. Aswe will discuss later, before the General Scholium, added to the second editionand refined for the third, there was nothing to satisfy this need for theorizing aboutsources. But there was already an “explanation” for this universal force—theCartesian plenum, and for its centripetal action—the vortex model, and it was thisNewton sought to quash with his analysis of motion of bodies in continuous media.The treatment is general, without specific reference to the cosmic medium.

Coming as it does after the full presentation of the mechanics of particlesand massive finite bodies, Book 2 of Principia was the explicit response to andrefutation of Descartes’ qualitative picture of the fluid-filled universe. If the fluidwere viscous, the moving masses would experience a drag that would change theirorbits over time. This was a negation, from an unusual direction, of the Cartesianviewpoint. A vortex could indeed explain the circulatory motions, although itwould apply only for circular orbits. An elliptical vortex is inexplicable, and wenow know not even stable. Newton also demonstrated that the velocity law fora central vortex could not produce the known law of planetary motion. To makematters even more difficult to arrange, nesting these was a nearly impossibletask. For a universe filled with such motions, each center was the source ofgravity.

There were several reasons for this frontal assault on the vortex picture. The firstwas philosophical. The Principia is really one long argument5 showing how theapplication of the concepts of inertia and force, along with the affirmation of theuniversality of gravitation acting on all masses in the universe, both explainsthe existing phenomena and predicts new ones. But there was a sub-text thatremained even more important: whatever gravity is isn’t important. This point wasdriven home in the General Scholium that Newton added to Book 3 when revisingthe book for the second edition. There he explains what had been only implied inBook 2, that it isn’t necessary to understand the ultimate causes (even though New-ton had his own ideas about them), it was enough to understand the mechanisms.The Cartesian picture was a qualitative, not quantitative, picture of how thingswork but it was also a proposal for what things are. The end of Book 2 contains anexplicit, biting scholium summing up the whole case against the vortex theory ofplanetary motion, ending with the statement:

Therefore the hypothesis of vortices can in no way be reconciled with astronomicalphenomena and serves less to clarify the celestial motions than to obscure them. Buthow these motions are performed in free space without vortices can be understoodfrom Book 1 and will now be shown more fully in Book 3 on the system of the world.


Figure 4.3: Figure from Descartes’ Principia showing hisexplanation for gravitation based on a universal fluid filledwith contiguous vortices. Each star is the center of a vortex.This is the world later described by Voltaire and contrastedwith the Newtonian vacuum. It was this model for planetarymotion that spurred Newton’s composition of the secondbook of his Principia. Image copyright History of ScienceCollections, University of Oklahoma Libraries.

An indication of the success of thesecond book is the remark by Huy-gens, originally among the most ded-icated Cartesians, that vortices disap-peared in the strong light of Newton’sdemonstration of their failure to pro-duce the Keplerian laws of planetarymotion. Most later commentators un-derstood the point as well. But therewere those for whom the Newtonian con-cept of gravitation didn’t suffice to pro-vide a complete, coherent philosophicalconstruction. One of them was BernardFontenelle and it’s worth a moment toreflect on his reaction to the Principiato see how strange the new concept offorce was to Newton’s contemporaries.

In his popularization of the Carte-sian system in 1686 (and many latereditions), Fontenelle extended the vor-tices to fill the universe, imagining eachstar the center of its own system with amultiplicity of inhabited worlds. By co-incidence, the work, Conversations onthe Plurality of Worlds, appeared nearlysimultaneously with the first edition ofthe Principia. The work was immenselypopular on the continent, more acces-sible than Descartes or, later, Newton,and written with a high literary style andand lightness of presentation. It was aserious work, and consequently also aserious challenge for the view that spaceis an empty immensity in which grav-ity acts by principle without a mechani-cal explanation. For the literate public,

Fontenelle was both a port of entry into the new philosophy and a guide through thenewly discovered territory. His popularizations of the activities of the Academiedes Sciences was the first yearbook of science, and was widely distributed and readon the Continent. Dedicated to this mechanical explanation for gravitation, hislast word on the topic was published anonymously in 1752 and again championedthe Cartesian system nearly 30 years after Newton’s death. There is nothing inthis picture that requires the vortices; it was only Fontenelle’s choice to have aphysical explanation behind the mechanism.

But there were other reasons for Newton’s digression on continuous media andfluid mechanics in Book 2. Remember that the pendulum was used for measur-ing not only time but also the local gravitational acceleration.6 Since Galileo’s


Figure 4.4: Isaac Newton. Image copyright History of Science Collections, University of Okla-homa Libraries.

discovery of the isochronism of the pendulum, a simple thread and bob provided astandard a fundamental measure of time at any place. But the comparison betweensites requires a very precise calibration and this can only be done knowing how totreat both the friction of air and the increase in the effective mass of the bob whenmoving through the background. This would later be indispensable in the mea-surement of the arc of latitude of the Earth and the study of its gravitational field.In addition, the motion of a body in air, instead of a vacuum, was a mechanicalproblem of some interest, especially for ballistics, and the complete solution forthe dynamical resistance of air was quite useful and an important extension of theproblems treated in the first book. It was another way of seeing what inertia meansbecause, since the frictional and resistive forces of a surrounding medium dependon the properties of that medium (for instance, its density and temperature), itsimpedance of the motion of a projectile dissipates the impulse. This had been seencompletely differently in medieval mechanics: in the impetus theory, you’ll recallthat the medium actively drives the motion that would otherwise decrease withincreasing distance from the source. In addition, Newton hypothesized that theforce of the medium depends on the velocity of the body, hence unlike gravitationor contact forces it decreases as the body slows down and is independent of thedistance it covers.


The System of the World

The first two books of the Principia were methodological and demonstrative. Thethird contained the applications, but not, as they had been for Galileo, to problemsof terrestrial mechanics. Newton had a much vaster territory in mind, the wholeuniverse. And to show the power of his new conception of gravitation nothingcould serve better than the explanation of some of the most perplexing naturalphenomena that had remained problems since antiquity. The tides, precession ofthe equinoxes, and the stability of orbital motion against perturbations were thesubjects of the third book of the Principia, the promised The System of the World.Here Newton dealt with the particulars, the demonstrations of the explanatory andtechnical power of this new physics: its application to the fundamental questionsof astronomy.

Try to imagine what it must have felt like, to see the whole of Creation opento analysis, to sense the evolution of everything, and to have a conceptual andcomputational battery with which to accomplish the task. A century later this wasthe inspiration for poets, as William Wordsworth wrote in his The Prelude (1805),

And from my pillow, looking forth by lightOf moon or favouring stars, I could beholdThe antechapel where the statue stoodOf Newton with his prism and silent face,The marble index of a mind for everVoyaging through strange seas of Thought, alone.

The tides provided one of the earliest successes for the Newtonians. Its clearestpresentation is, however, not Newton’s own but the 1696 memoire by Halley forthe Royal Society. Newton showed (Cor. 2) that the center of gravity—(i.e. centerof mass, here gravity has the same meaning as it would have to Galileo)—moveswith a fixed angular momentum and in a closed orbit, experiencing only thecentripetal force of the attracting mass. But if we have an extended ensembleof bodies, which are mutually attracting and also attracted to the central mass,those at larger distance cannot have the same orbital period as those closer tothe center. Since they are forced to move with the same velocity as the center ofmass at distance r from the central body, in the orbiting frame their centrifugalacceleration compared to the gravitational attraction of M produces a differentialacceleration. The body will distend along the line of centers to balance this excesswhile keeping the same motion of the center of mass. The oceans remain in overallhydrostatic balance (this was the second time hydraulics was invoked on so largea scale, the first was Newton’s argument about the equilibrium of channels andthe shape of the Earth in Book II) but rise by a small amount in instantaneousresponse to the change in this differential acceleration. The solid Earth transportsthe fluid and the static response, independent of any induced flows, causes a twicedaily, or semi-diurnal, change. The bulge of the fluid is coupled to the rotation ofthe Earth and this mass produces an equal and opposite reaction (again, the roleof the third law) on the perturber. In this case, the Sun is so much more massivethat the effect on its structure and motion is insignificant.


For the Moon, the situation is complicated by the difference between the periodsof lunar orbital motion and terrestrial rotation. The Moon moves about 1/30 of itsorbit per revolution of the Earth, in the same direction, so it always leads the Earthwere the Earth to stay motionless. But since the rotation period is one day—justsaying this the other way around—the bulge is always carried forward and thisextra mass is now always ahead of the instantaneous position of the Moon. Beingso low mass, the Moon receives a greater relative acceleration than the Sun andmoves outward in its orbit. The Earth, as a reaction, spins down. This producestwo effects, a steady lengthening of the solar day and a lengthening of the periodof the Moon’s orbit.

THE GENERAL SCHOLIUM

On the urging of Roger Cotes, his editor for the second edition of the Principia,Newton was induced to add something about his theology to counter charges ofimplicit atheism in the work. It is hard to see now why such an orthodoxy wasrequired. The statements in many of Newton’s writings, such as his correspondencewith Richard Bentley regarding the implications of the new physics for religion,certainly attest to a deep conviction in the direct action of God in the world. Butsomehow the system was incomplete and through this qualitative and extremelycarefully worded gloss on the whole of the work, Newton also touched on one ofthe fundamental cosmological questions: the stability of the system.7

Hitherto we have explain’d the phaenomena of the heavens and of our sea, bythe power of Gravity, but have not yet assign’d the cause of this power. This iscertain, that it must proceed from a cause that penetrates to the very centers of theSun and Planets, without suffering the least diminution of its force; that operates,not according to the quantity of surfaces of the particles upon which it acts, (asmechanical causes use to do,) but according to the quantity of the solid matterwhich they contain, and propagates its virtue on all sides, to immense distances,decreasing always in the duplicate proportion of the distances. Gravitation towardsthe Sun, is made up out of the gravitations towards the several particles of which thebody of the Sun is compos’d; and in receding from the Sun, decreases accurately inthe duplicate proportion of the distances, as far as the orb of Saturn, as evidentlyappears from the quiescence of the aphelions of the Planets; nay, and even tothe remotest aphelions of the Comets, if those aphelions are also quiescent. [Buthitherto I have not been able to discover the cause of those propertiesof gravity from phaenomena, and I frame no hypotheses. For whateveris not deduc’d from the phaenomena, is to be called an hypothesis;and hypotheses, whether metaphysical or physical, whether of occultqualities or mechanical, have no place in experimental philosophy. Inthis philosophy particular propositions are inferr’d from the phaenomena,and afterwards render’d general by induction. Thus it was that theimpenetrability, the mobility, and the impulsive force of bodies, and thelaws of motion and of gravitation, were discovered. And to us it is enough,that gravity does really exist, and act according to the laws which wehave explained, and abundantly serves to account for all the motions ofthe celestial bodies, and of our sea.]


And now we might add something concerning a certain most subtle Spirit, whichpervades and lies hid in all gross bodies; by the force and action of which Spirit,the particles of bodies mutually attract one another at near distances, and cohere,if contiguous; and electric bodies operate to greater distances, as well repelling asattracting the neighbouring corpuscles; and light is emitted, reflected, refracted,inflected, and heats bodies; and all sensation is excited, and the members of animalbodies move at the command of the will, namely, by the vibrations of this Spirit,mutually propagated along the solid filaments of the nerves, from the outward organsof sense to the brain, and from the brain into the muscles. But these are thingsthat cannot be explain’d in few words, nor are we furnish’d with that sufficiency ofexperiments which is required to an accurate determination and demonstration ofthe laws by which this electric and elastic spirit operates.

Thus, in the General Scholium, although he took great pains to make his philo-sophical (more to the point, theological) position clear, Newton flatly refused—inany edition of the Principia—to enter into a debate on the cause of gravity, as-serting that it is merely an intrinsic property of matter. This was not his problem,nor did he recognize any need to seek its cause. Although he seemed to almostecho an unlikely predecessor, Ptolemy, who considered that the aim of a modelis to reproduce the appearances of the heavens—to account for their motions—Newton’s physics went much farther. He required complete consistency amongall parts of the explanations of the motions and only avoided questions of ulti-mate causes. Yet he couldn’t completely avoid the issue, not only because of hisaudience’s expectations but also because of his own theological preoccupations.The Prime Mover had been eliminated as the source for the motion but God wasnow required to maintain the system in a state of perpetual motion and stability.So in the General Scholium Newton included the seemingly enigmatic declaration“hypothesis non fingo,” I frame no hypotheses but the context of this affirmationwas unambiguous. He lacked any causal explanation for gravitation, postulating itas a universal force that acted between bodies only by virtue of their mass. In this,his construction of a physical picture is completely concordant with the modernprogram of physical science. By postulating an attribute and providing a formalmathematical representation of its force law, the direct problem of motion can besolved within a consistent set of axioms, derived from experience. The details ofthe interactions, how the force is transferred between bodies for instance, doesn’tmatter as long as the mathematical law for its action can be prescribed. There isno need to postulate final causes in the original sense of that word, the ultimateorigin of these physical laws, to make progress.

The universe becomes spatially unbounded in Newton’s final conception. Inorder to preserve its stability, he demonstrates the enormous distances of the starsand that their cumulative forces should be very small. Homogeneity of the overalldistribution is required, any excess would become self-gravitating and attractiveto all masses in its immediate vicinity. Given enough time, this should lead to acomplete collapse of the entire system.8 His provisional solution was theological,that the system is occasionally “reset” from the outside by God to maintain thestate of the universe.

In summary, even without a cause of gravity, the inheritance of the Newto-nian revolution was an intoxicating the vision of a truly self-consistent natural


philosophy that could explain all phenomena. The successes in celestial me-chanics, in fluid dynamics, and in the gas laws were truly impressive. But whilemechanics was reformed by the Newtonian methodology, it took almost two hun-dred years to explore its richness and firm up its foundations.

THE NEW CELESTIAL MECHANICS

Planetary motion was the inspiration for, and critical test of, Newtonian gravitationtheory. Solving the problem of the orbit for two isolated bodies, the basic Keplerproblem, was sufficient to demonstrate the success of the new concept of a universalforce acting between masses at a distance but not enough to extend that theory tothe construction of the world. The success of the Newtonian concept of force andthe inertia of circular motion removed the need for tangential drivers, which you’llremember even Kepler required, and provided the means to geometrically modeland calculate the actions of forces in arbitrary arrangements of masses. we’veseen, since an acceleration requires a force as its cause, and the accelerations arethe observables in celestial mechanics in ways more obvious than anything elsein Nature, the forces can, in principle, be deduced by a physical procedure. Butwhat does that mean? When you observe a planet’s position, it is with respectto the stars whose positions provide a fixed reference frame. The lunar theorywas the first success of the new celestial mechanics so let’s examine how thatevolved as a way to understand this statement. The month is the mean motionof the Moon against the stars, the circular orbit that should result from motionaround a gravitating center of mass. But this constant period doesn’t suffice. Therate of motion varies with the position in the orbit and the season of the year. Apurely kinematic approach takes these and assigns each a “correction,” withoutexplaining the origin, so you have at the end a complex set of rules. These can besimply encoded as a set of tables that depend on a number of variables: the timeof the month, the time of the year, or the phase in the motion relative to some fixedpoint. These are the deviations. With the Keplerian ellipse the most important areimmediately accounted for, since the motion is not actually circular the effect isto alter the angular displacement depending on the phase in the orbit relative tothe reference position independent of the time. There is a choice here, one canuse the orbital phase or the time after perigee passage, they’re equivalent. Thesame construction, with less success, had been attempted by the introduction ofthe eccentric in the geocentric picture for the Sun and planets, for the Moon itwas in a sense simpler because it is really going around the Earth. The differencecomes when attempting to predict phenomena such as phases of the illuminationor eclipses. Then you need to solve the motion relative to a moving object, theSun, and this requires a second orbit. It was this which produced the difficultiessince the phase of the perigee, relative to the solar motion changes in time. Thusthe time, not only the angle in the orbit, is required. It is still possible to refereverything to a circular orbit, however, or to a Keplerian ellipse, and treat eachsuccessive alteration purely geometrically to obtain the successive approximationsto the exact motion and predict the times of eclipses and occultations as well asphases.


Even by the middle of the seventeenth century it was not clear that the changesin speed required a force and that agent could depend on the distance. This isperhaps best seen in Galileo’s misapprehension of the cause of the tides. Theproblem of freefall is fundamentally different in appearance than that of planetarymotion because it doesn’t require a cause for the acceleration, at least not a preciselaw. The law for gravitation was not Galileo’s concern. He was dealing with forcesof a different sort, and especially the general notion of inertia and equilibriumof bodies under mutual action. The rest would remain speculation. What keptthe Medicean stars moving around Jupiter, or cause the companions of Saturnto move with the planet, was not his question, that it happened was enough todemonstrate that a center of circular motion could itself move. Thus the Mooncould stay attached, somehow, to the Earth while it orbits the Sun. Within theNewtonian approach, however, any change in the angle relative to the centralforce orbit, the reference ellipse, requires an additional perturbation. All changesin motion are along the lines of the forces, thus all orbits require centripetalforces. The proposition thus allows one to discover the origin of every irregularityby determining the sense, at any instant, of the change in the state of motion. Thelinear superposition of the forces in the same direction and parallelogram law fortheir addition if they are not parallel or even collinear, both demonstrated in BookI of the Principia, suffices to insure that once an acceleration is known so is themagnitude and direction of the force that produces it. Thus we see the tremendouspower of the laws of motion as Newton stated them in the celestial context. Thefirst states “look at the motion, see if it is changing in direction and/or magnitude;if it is, then there is an external cause.” The second continues “and, by knowingthe acceleration you know the direction of the cause,” finishing with the third lawasserting “and the magnitude of the mass multiplied by the acceleration is directlyequal, for any quantity of mass and irrespective of its composition, to the forceproducing the change.” The inverse problem can be solved once you can makea kinematic measurement, to reason from effects back to causes without needingadditional hypothetical constructions.

These variations are accelerations caused by forces that act along the linesof centers to the other bodies. Thus every variation from a perfectly centrallysymmetric motion requires an additional force “agent” or property of the body,just a different center of attraction. For instance, the solar mass is large enough torequire including its attraction in the motion of the Moon even though the distanceis enormous, and this is one of the causes for the orbital precession and also of thetides. If it causes one because of the finite size of the orbit of the Moon, it producesthe other in a similar way by the finite size of the Earth. One unique explanationsuffices for a host of apparently distinct phenomena. The additional proof that thegravitational attraction is central only if the body is spherical and that there canbe additional accelerations produced by a finite, nonspherical body, extended theforce law to treat much more complex problems, eventually even the motion ofsatellites in the near field of a nonspherical planet.

Remember that Newton never, throughout the long discussion in Book I, spec-ified what gravitation is. How it is transmitted between bodies was left to specu-lation, of which there were many in the subsequent years. He faced the criticismof the Cartesians who, even in the absence of quantitative results, propounded a


physical origin for the field that Leibniz called an occult quality —attraction—thatwas universal and external to the body. Whatever his conceptualization of its me-chanical origin, Newton’s program was to demonstrate how it unified and resolvedfundamental problems and rendered calculable things that had previously beentreated only phenomenologically. Even if he didn’t solve many of e problems heposed, in particular those of most importance to the calculation of the lunar orbitsuch as the variations in latitude and the tides, Newton succeeded in setting outthe framework in which such problems could be solved with eventual developmentof the computational machinery. These results had a dramatic effect on Newton’scontemporaries and framed the research program in celestial mechanics for nearlytwo centuries until the possibility of non-gravitational forces was introduced at theend of the nineteenth century. Again, we can see why. The single source for theaccelerations reduced the physical complexity of the problem. Only the form ofthe law, not its ultimate origin, is needed along with the equations of motion. Theconservation laws reduce to only that of inertia. There is no need to separatelytreat angular and radial momentum, a tangential force resulting from the presenceof another body than the principal center of attraction or any nonspherical centralor external mass distribution will produce a change that is identical to a torque.In the two body problem this is the simple explanation for the stability of thedirection of the line of apsides (if there is nothing to break central symmetry anorbit once fixed in space remains in the same orientation) and the constancy ofthe eccentricity. When in Book II Newton digresses at length on the dynamicsand properties of fluids, and in the third book asserts that vortices are completelyunnecessary to explain the planetary motions, he understood that in a vacuumthere is no drag (hence the linear momentum remains constant subject only togravitational driving) and that the tangential motions were imposed at the start ofthe world and thereafter required no additional force to maintain them.

The first predictive, as opposed to explanatory, success of the new mechanicswas the return of the comet of 1682, the comet that initiated the exchange withHalley that finally impelled Newton to compose more than the short tract de Motuand produced the Principia. It was also the demonstration of the universality ofgravitation. Comets presented a particular difficulty for astronomers. They hadbeen included in Aristotle’s Meterology as terrestrial phenomena, notwithstandingtheir affinity to the heavens because of their diurnal motion with the stars. Buttheir appearances were random, both in timing and location, and their motionswere never as regular as those of the planets. Furthermore, they are obviouslyvery different bodies, nebulous and irregular, moving both within and far fromthe ecliptic. With Georg von Peuerbach’s failure, in 1457, to detect a cometaryparallax and Tycho Brahe’s solution for the orbit of the comet of 1577 that requireda radial motion that crossed the spheres, these bodies were obviously beyond theatmosphere. Descartes explained their paths as complicated weaving betweenplanetary vortices, and Galileo and Kepler had also dealt with their appearancesand motions. But with the apparition of an especially bright new comet in 1682,Halley noted the regularity of historical comets with intervals of about 76 yearsand hit on the possibility that this was the same body as that seen in 1607 andearlier. Not only was it superlunary but it was also recurrent. Armed ultimatelywith the machinery of the new gravitational theory, he determined its orbit as a


conic section, an ellipse, and predicted its the return in 1757. When first detectedon 25 December 1758, it proved a sensational triumph for the new mechanicsand spawned a mania to search for, and determine the properties of, new comets.Since the orbit is very elliptical, and the aphelion distance is so large that itreaches beyond the then-known planets, the orbit itself extended the reach of testsof gravitation to about 30 times the distance of the Earth from the Sun, past theregion covered by the theory presented by Newton.

Similarly, a cosmical test for gravitation came with the determination of therelative masses and densities of the planets. It was possible as early as 1748 forMaclaurin to present a table of the relative constitution and properties of the majorbodies in the solar system, Jupiter and Saturn on account of their moons and theSun by the period of Mercury scaled to that of the Moon, in terms of Earth masses.Thus the solar perturbing contribution to the lunar orbit could be establishedand with that, the motion of the Moon could be predicted with fair accuracy.With this determination of the masses of Jupiter and Saturn and a knowledge oftheir reference orbits, the anomalies in their motions were also explicable. Oneof the first challenges was to solve the motion of these planets under their mutualperturbation. In a prize submission to the Paris Academie in 1743, Leonard Eulershowed that the motion is stable under periodic perturbations This became knownas the Great Inequality, the near resonance of the two orbits in a 5:2 ratio of periods(Saturn to Jupiter). The near resonance means that with the passage of time, theforce becomes quite large because the perturbation remains almost stationary inthe moving frame and never cancel on average. Any resonance (integer ratio of theperiods) causes serious, potentially fatally disruptive problems for orbital motionunder any sort of field, or even a harmonic oscillator.

The three body problem isn’t merely the addition of another mass, it requiressolving the equations of motion in a rotating frame. Formulated to address thelunar problem, in which the principal masses (the Earth and Sun) have an orbitalperiod of one year and small eccentricity, the problem was only made tractablewith mathematical advances and not resolved fully until the twentieth century.But in the mid-eighteenth century, it had become a serious challenge to thestability of the world. The fundamental problem was set out in Book I of Principiaand elaborated in Book 3. Pierre-Simon Laplace (1749–1827) finally succeededin showing that the perturbations were periodic on a very long timescale and,therefore, although the motion might appear now to be changing in only one sense(that is, secular), over a greater interval of time the orbits are stable. Lagrangedemonstrated that there are a set of fixed points in the orbit of a third body ofnegligible mass (relative to the principal components) where perturbations leavethe mass unaltered. In the so-called restricted problem, in which the eccentricityof the main binary is vanishingly small and the orbital period is constant, thereare two equilibrium points (where the attractive force is balanced by the inertia ofthe particle through the centrifugal force that appears in the non-inertial rotatingframe. These lie at the vertices of equilateral triangles and are along the lineperpendicular to the line of centers of the binary on opposite sides of the orbit butin the same plane. There are three more, where the force again locally vanishes,but these are unstable and lie along the line of the principal binary. The discoveryof the Trojan asteroids, at 60 degrees relative to the Sun in the orbit of Jupiterdemonstrated the existence of the L4 and L5 points.


The discovery of Neptune in 1846 was the most celebrate result of celestialmechanics of the nineteenth century.9 Following its discovery, Uranus had nearlycompleted an orbit at the end of the 1830s by which time the secular departurefrom its predicted path was significant. While this could be merely the result ofaccumulated bad positions, the astrometry was sufficiently precise to afford analternative, and more plausible, explanation: the orbit was being gravitationallyperturbed by another, yet unknown, planet. By that time, the masses of the principalplanets are known from their moon systems. Since each perturbation produces notjust a displacement but an acceleration (the angular motion changes in time), inthe now standard procedure of the inverse problem the location of the perturbercould be obtained by noting the time dependence of the changes in Uranus’motion. If the perturber is stationary, the reference orbit won’t close and precesses.Instead, if the perturber is moving, the axis between the bodies will follow themotion of the unknown orbiter. The equations of motion, though nonlinear andtherefore almost impossible to solve in closed analytic form except in very specialapproximations, can nonetheless be integrated numerically since they dependonly on the initial conditions. Two solutions were found, by John Couch Adams(1819–1892) and Urbain Jean Joseph Le Verrier (1811–1877), both similar, butthe latter was the basis of the discovery of the planet by Galle at the Berlinobservatory on 23 September 1846. In the most stringent test, on a cosmic scale,celestial mechanics was thus seen to have spectacularly confirmed and firmlyestablished the description of force in general, and gravitation in particular, as ithad developed during the previous 150 years. Throughout the nineteenth centuryan increasingly accurate and complex set of algorithms were developed for justthis problem, how to find the future orbits of the bodies of the solar system knowingonly their current parameters. The calculations of orbits was a sort of industry inthe nineteenth and early twentieth centuries, requiring patience and long tediousnumerical manipulations by hand. The inventions of the differential analyzer (byVannaver Bush in the 1930s) and later appearance of electronic computers (duringthe World War II) made possible in minutes or hours computations that had takenmonths or even years to complete.

ACTION, VIS VIVA, AND MOTION

In an age of “universalists,” Gottfried Leibniz (1646–1717) stands out as one of themasters. Distinguished in almost all areas of philosophy, he was also the node of anetwork of savants through scientific societies and his journal, Acta Eriuditorum,and among his other activities independently developed the calculus based onintegration rather than differentiation and created one of the first mechanicalcalculators. His fierce rivalry with Newton makes for a gossipy history as a clashof personalities and the social assignment of credit for discovery. But that’s anotherstory. The development of mechanics required many minds and many approaches.Once the Principia appeared its elaboration by the Continental mathematicians,inspired by the Cartesian program that Newton sought to replace, produced a verydifferent foundation for forces and mechanics.

Leibniz’s objected to Newton’s invocation of “attraction” as a property of matterand, in general, to the “un-mechanical” approach of reasoning without firm causalfoundations. The issue came to a head in the extended correspondence between


Leibniz and Samuel Clarke, the last editor and amanuensis of Newton. In a seriesof letters exchanged over a period of several years and finally published forpublic consumption, Clarke—acting as Newton’s voice at the latter’s direction—defended the methodology of the Principia. The core of his argument is that wecan observe an effect and infer an action even without knowing the details of howeither occurs. If we see a body moving in a curved trajectory, recall by the firstlaw, this demands that something not only deviated but continues to deviate themass. If we know that the “something” is a force, and we know somehow whatthe dependence of that force is on the masses and distances, we can determinethe trajectory without knowing what the agent “is.” Further investigation mayyield a deeper understanding but the formalism suffices when the questions areabout what will happen rather than why. The latter is left to speculation, or whatNewton derided as “hypotheses.” Leibniz distinguished between two “forces” incontrast to Newton. One, vis muotor (moving force), he argued, is responsiblefor the production of motion (the extension of the Aristotelian principle and theNewtonian construction). The other, the “living force” or vis viva, is an attributeof the body while in motion. The fall of a body yields work because the externalforce, gravity (not distinguished from weight) accelerates the mass and producesan impulse when the body strikes the ground. The quantity 1

2 M V 2 is the kineticproduct of this work and this is the vis viva. While Leibniz did not include the factorof 1/2 (this was realized only later by Kelvin and Tait when writing their Principlesof Natural Philosophy in the mid-nineteenth century), this doesn’t matter for thegeneral purposes of the argument. He posited that the quantity vis viva timesinterval of time for the motion is the action of the external force. This should bea minimum for the path chosen by the mass since a straight line is the shortestdistance between two points, for a unidirectional constant acceleration comingfrom its potential to fall because it is lifted. This almost resembles natural place,and even more the principle of work from Jordanus, but it is a subtler concept.

Imagine we lift a weight to a height and then release it. At the ground, theimpulse will be greater for increasing height, of course, but so will the velocity.No matter what height the body falls from, it will always reach a final speed thatdepends only on the height. We can, therefore, say that the quantity that measuresthe effect of the weight is not the impulse but the quantity that measures themotion, the vis viva. The difference between this and Galileo’s conception is nottrivial. While the impact of the body is certainly greater, this can be lessened byimagining the body to be compressible. On the other hand, for the vis viva, we arenot imagining its effect on the surface at the end of its motion. Instead, we askwhat it does during the fall, and how it accelerates. The longer the fall, the longerthe acceleration, but the amount of vis viva gained will always precisely the effectof the vis mortis.

We can see this dimensionally, something that requires that we know that aforce is the change in the momentum with time. The work, or potential for motion,is the product of force with displacement and thus varies as (M L/t2) × L. Butthe velocity has the dimensions L/t so the work is the produce of M V 2t , whichis the action. The centrifugal force is Mv2/L for displacement around a curvewith length scale L so the work again has the same dimensions as the vis viva.Leibniz’s hypothesis was that the action is always minimal along the trajectory for


an accelerating body acting under the influence of a “dead” force (in the case offreefall, its weight). This is the first statement of the mechanical principle of leastaction to which we will return later in the book. This quantity was introduced intophysical reasoning at an intermediate approach between Galilean and Newtoniandynamics when the distinction between weight and mass was still unclear. Leibnizchose, by happy coincidence, the one case where the distinction isn’t necessary inclassical physics. Since all bodies fall at the same acceleration regardless of theirmass, for the purpose of defining action we can equate mass and weight. Second,the definition involves an integral, which means a sum, and leaving aside problemswith the definition of the integral that would not be cleared up for several centuries,there is no need to think in terms of instantaneous quantities such as the speed atany moment. Finally, this is a quantity that has no direction; it is an attribute ofthe falling body along the trajectory, whatever the trajectory is, so we don’t needto think about the reaction to an imposed force in any specific direction. It’s atotal quantity in the sense that it belongs to the moving body. Recall that Galileo,through his experiments with inclined planes, was able to separate the inertialcomponent of the motion, the horizontal speed, from its vertical, accelerating partand thereby note that the vertical component alone determined the restitution of thebody to its original height at rest, frictional and resistive effects notwithstanding.Leibniz took this a step further and allowed for the possibility that some of the visviva would eventually transform into compression for elastic bodies.

Although the language is anachronistic, we can say that the amount of energyis conserved by a moving body, whatever its form, as is the impulse or momentum.This permits us to treat collisions as well as motion under constant forces. Foran impact not only accelerates a body from rest and imparts a momentum, italso gives the body a vis viva that is transferred from the collider to the target.We can then say, as Leibniz did, that the total momentum and the total vis vivaremain with the movers throughout time unless something else brings the bodiesto rest. One, momentum, acts along the direction of the collision. The other isindependent of direction. This is consistent also with Descartes’ concept that thequantity of motion in the universe is constant, just redistributed among bodies asthey interact. The new approach opened a vast territory to mechanicians who nowcould apply a small number of physical principles to a huge range of problemswithout feeling constrained to explain the origins of the forces involved. Instead ofhypothesizing a mechanism and seeking an effect, experiments and observationswere now the tools for understanding the general properties of matter by rigorouslyreasoning from effects back to causes.

ELASTICITY AND FLUIDITY

The new machinery of the calculus and the new laws of mechanics were appliedto more than just theoretical questions. They had a very practical consequence,engineering. While Galileo had begun the reform of statics, now problems ofstability and response to changing conditions could be addressed. During theeighteenth century, the interest in dynamical reactions of objects to distortions andloading increased as technological developments required better understanding ofthe strength and behavior of materials. As projects grew in scope and complexity,


Figure 4.5: Leonhard Euler. Imagecopyright History of Science Collec-tions, University of Oklahoma Libra-ries.

and with the definition of force allowing quantification ofthe conditions governing machines and structures, simi-larities were seen between what had previously appearedto be very different problems.

Leonard Euler (1707–1783) was among the first toapply the new perspective to the dynamics of bars andplates, leading to a general theory of material elasticity.Let’s look at how it was accomplished. A bar can beviewed as a collection of strings (imagine the analogy be-tween a coaxial cable and a cylindrical bar, for example).Then if one end is clamped and the other end is loadedwith a weight, the bar bends. When Galileo treated thisproblem, you know he considered only the breaking pointof the load and the scaling laws for the strength of thebeam. On the other hand, we might want to know whatvibrations would be possible for this object, dependingon how and where it is clamped.

Elastic media provided a superb challenge for Eu-ler and his contemporaries for applying force conceptsbeyond collisions and trajectories. It also required a mod-ification of the concept of inertia. For a point, this is thetotal mass. But for a string, or a membrane, or a prismor beam, each tiny piece of the body is interconnected

with the others and the calculus becomes indispensable. The matter is distributedcontinuously with some density and the force is also distributed over the volume.A string or spring is an ideal example. Hooke had considered only equilibrium.Now, having the relation between force applied and the reaction produced, it waspossible to ask questions about what happens if this equilibrium is disturbed.Pulling the string induces a distributed reaction, the tension, and the strain de-pends on position. Its density may also depend on position, the mass per unitlength may not be constant (depending on how the string was made). If you releaseor reduce the applied force the string moves to, and then continues past, the equi-librium. If it compresses, there’s a contrary reaction that produces an extension,on expansion the sign of this reaction reverses, and the resulting motion is cyclic.For small deformations, any elastic material behaves according to Hooke’s lawand the tension is linearly proportional to the strain with the constant of propor-tionality depending on the properties of the material, the Hooke constant. Sincethe reaction is a force, there is a characteristic timescale for the oscillation thatdepends only on the length of the string (its total mass) and the strength of thereaction. In other words, we can compute the frequency of the vibration and thusobtain the string’s elastic properties by looking only at the oscillation. Then wecan extend this to two or three dimensional things, such as sheets and beams, byimagining them to be composed of networks of masses with string-like intercon-nectedness.

In both cases, we have a harmonic oscillator, one that has a fixed frequencythat is independent of the amplitude of the vibration and only depends on the sizeof the equilibrium system (in this case, the elastic constant, mass or moment of


Weight

Figure 4.6: The bending of a beam showing the differential strain and stressdescribed by the two-dimensional application of Hooke’s law of elasticity. Ifthe weight is released, the beam will oscillate above and below the horizontal.

inertia around the point of oscillation, and total length). The most familiar form ofNewton’s second law, F = ma, is due to Euler and serves when the mass (inertia)is constant for the mover. Then for a spring, the one dimensional change in lengthproduces a reaction of equal magnitude and in the opposite direction dependingonly on the extendibility of the material—the constant of proportionality found byHooke, K . When there are no other forces acting other than the spring itself, themotion is periodic and independent of the amount of stretching. The frequencydepends only on the elasticity and the inertia of the spring plus the attached massso a stiff spring has a shorter period for any mass, and a heavy load oscillates withlonger period. We can generalize this to the case of a continuous mass loading,that the spring is moving under its own weight for instance, and then we have thesame case as a pendulum with a moment of inertia. If the spring hangs freely,without loading, its mass produces a variable stretching that depends linearly ondistance but if disturbed the oscillation has this characteristic frequency.

We can again make an appeal to dimensional analysis. We have the three lawsof motion, and the basic units of mass, length, and time. Call these M , L, and t .In these units, the change in a length dimensionally L. Thus Hooke’s law can berecast, dimensionally, as M L/t2 ∼ K L, so the units of the constant K are M/t2.But this is the same as a mass times the square of a frequency since the unit offrequency (now called Hertz or Hz) is ω ∼ 1/t . Then the frequency of oscillation ofa string, ω, for which the Hooke law holds precisely in the small deformation limit,scales as ω ∼ √

(K/M ) for a string of mass M and the dispersion relation—theconnection between the frequency and the wavelength of the vibrator—becomesω2 ∼ (K/M )L−2. Thus the higher the tension, the higher the frequency, and thehigher the mass the lower the frequency and this gives a wave that moves alongthe string. To extend this to a beam requires changing to a stress but then themoment of inertia of the beam enters and the frequency scales as (Y /I )1/2 wherewe take Y , the Young modulus, to extend Hooke’s law to the finite width of thebeam. Now since the dimensions of Y /I ∼ L−4 we have for the dispersion relationω2 ∼ (Y /I )L−4. This is different from our previous result for the string becausewe need to include the mass per unit area and the force per unit area (hence the


L−4 dependence). A beam has a variable radius of curvature that depends on itsthickness (it’s a case of a differential force, or shear).

The important step Euler took was to extend this to the more general case ofdisplacement in two or three dimensions. A string, is a springy line. Plucking it, theforce depends on the change in the length of the string because the displacementsare directed along the perpendicular direction. For a symmetric displacement,Euler could write the force in terms of the difference in the slopes of the cordat each point along its length, x, and then take this as a second derivative withrespect to that coordinate which yields the same dispersion relation we foundusing dimensions alone. This also provides a very important insight into themeaning of inertia, one almost anticipated by Buridan in his Questions on thePhysics. Recall his statement that the string is a continually oscillating bodybecause it is constantly re-traversing its equilibrium once it’s gained an impetusby a violent motion. I say almost because the foreshadowing is really illusory.Buridan had no idea of inertia and that’s why the string passes its stable point:once the unbalanced force accelerates the distorted string toward the midpoint,when the force vanishes the motion doesn’t stop. How different this is from theAristotelian force law! You see here why the idea of mass was so critical to the finalformulation of a proper dynamical principle. With the requirement that motionrequires unbalanced forces, it is obvious the string should come suddenly to adead stop at the mid-place. But when the acceleration vanishes, the momentumdoesn’t and the passage of the string through the midplane reverses the sign of thereaction. Thus the string will again distend as much as it had originally, ignoringdissipation, and finally will reverse motion since again the forces are maximallyunbalanced at the extreme of the oscillation.

For a linear displacement, Hooke’s law suffices to describe the reaction force.But there are other ways to deform an elastic solid and this requires a morecomplicated theoretical machinery. Although for a wound spring it would seemyou need the notion of a twist, this wasn’t included. Instead, a spring—for instance,in a watch—was assumed to be locally extended and the force was caused onlyby the expansion or contraction of the length. But this isn’t the same thing asthe twist of a filament. Imagine a wire that is pinned at its two ends. Applying atorque at only one point produces a helix whose tilt angle changes as the wire istwisted further. In the limit of small deformation, the reaction is described usingonly the angle δφ and now with the moment of inertia instead of the mass, theoscillation frequency is similar ω2 = E/I . The coefficient is the same as for abeam, and emphasizes why the study of strength of materials is so connectedwith the Young modulus. Shear and twist are really the same thing, they’re bothnonaxial, torsional, distortions. This would later provide the necessary tool forthe measurement of the force law for charges and the gravitational constant (byCoulomb and Cavendish, see Chapter 5). As Galileo had discussed in the Two NewSciences, there is also a non-elastic limit for materials, a point beyond which thebody remains permanently deformed or even breaks. Euler’s treatment permittedthe introduction of higher order terms in the force law, which produce anharmonicmotions, and permitted the calculation of the critical limits to the distortions.

Euler and Lagrange were also, at the time, engaged in fundamental problemsof mechanics so it was natural that their thinking would combine their variousinterests when it looked as if the same techniques could be applied. So when


thinking about membranes and plates, they were also working on fields and fluidsand the inevitable cross fertilization was very fruitful. Particle dynamics didn’trequire a continuous medium but such questions of flexible response did. Theextension this treatment to the distortion of clamped beams in equilibrium tobeams in oscillation was not as simple but became the foundation of quantitativeengineering mechanics and was again posed in the context of least action. Imaginethe beam again to be a collection of fibers each of which is stretched by a length lδθ .The bending of the beam produces a differential stretching that does work throughelasticity. The Young modulus, E , and the moment of inertia I are required toaccount for the finite cross section. The bending is differential so there is a secondorder effect that is absent for a string or thin membrane but there for beams andplates: the differential stretching of the medium across the area transverse tothe axis of the beam or plate produces a second degree rather than first degreedependence.

This period also saw the development of a new science, acoustics, based on theideas derived from statics and small perturbations of elastic bodies. Newton hadtreated the speed of waves in continuous media, including the compression of airand the motion of waves in channels, but the interaction between vibrating sourcesand their surrounding media is a much more complex problem. It’s one of boundaryconditions. For instance, if a plate is in equilibrium under its own weight, staticstells you how it deforms. But if a rigid plate is set into vibration, for instance ifyou hear a gong, the vibration of the plate is being coupled to the surroundingair, producing a wave that carries energy outward from the plate. Once againthe problem becomes partly geometric, the symmetries of the plate determine itsvibrational modes. A membrane is the extension of a string to two dimensions sincethe moment of inertia doesn’t matter. Hence we can go to two dimensions withthe combined motions being only in the direction perpendicular to the plane. Thisacts like a piston on the surrounding medium which has its own elastic properties(this also worked for fluids in general). The local oscillation in one dimensionbecomes a forcing in three that begins at the surface and extends through thewhole volume. This is a wave and the solution of the equation that describesthe motion of the driving body, the boundary conditions, and the symmetry ofthe medium and the surface, since we could also write this in cylindrical orspherical coordinates depending on the shape of the oscillator. This was eventuallydemonstrated experimentally by E. F. F. Chladni (1758–1827) who studied thevibration of plates and membranes driven by stroking the edges in different placeswith a violin bow. He rendered the vibrations visible with a remarkably simpletechnique: knowing that the surface would consist of nodes and extrema of theoscillations, he reasoned that the nodal pattern would be stable to a steady stroking.Then sprinkling a fine powder or sand on the surface would lead to accumulationof the grains at the stationary points of the surface, avoiding those parts that arein constant motion. The method showed how to study the harmonic motions of anysurfaces, even those with shapes far more complicated than could be computed.10

There are engineering implications of this elastic force law since a structure mayhave resonances, depending on its elastic and inertial properties, that can causecatastrophic failure. This is because of the natural frequencies of the structure. Ifthe driver happens to coincide with some rational multiple of these frequencies,unbounded amplification can occur limited by the point of inelastic response


of the vibrator. An important application came after the invention of the steamlocomotive and the expansion of railroads, the vibrations of tressel structures andarches became an important area for practical mechanics within a century of thesefirst applications of continuum mechanics.

But there’s an even more dramatic application. Imagine you’re sitting in aquiet room and suddenly the floor begins to vibrate. The motion may last foronly a moment but it is an unmistakable experience. The building may sway,lamps may swing, dishes and pictures may fall, and the walls may crack. This iswhat it’s like to be in an earthquake, something that is familiar in many placeson Earth. Earthquakes were variously explained by invoking only local effectswithout considering the response of the planet as a whole. Quite naturally, theexplanations were of the same sort as for volcanic explosions: changes in thesub-terranian environment that caused upheavals of fluid and expulsion of gas.The local motion of the ground was in response to this. The seismograph, atleast the ability to measure the moment and direction of ground motion, appearsto have been invented in China in the second century by Chang Heng. Using asimple dropped pellet, the device registered the direction of the ground motion butgave little indication of its amplitude. This is just the opposite case to Chladni’svisualization method, the driving of a mass-loaded spring by the ground motionleads to a visualization on a chart of the oscillation of the spring. There isn’t justone frequency, the motion may have a very complicated spectrum, but elasticitytheory can treat both the local and global motion of the Earth.

FRICTION

Aristotle’s law required a driving force to maintain motion against resistancewithout identifying what either force is. In freefall, this was presumed to be theair since it has weight. But for motion along a surface this was less obvious. Thinkof a cube being dragged across a table. The friction opposes the motion but it’salong only one face of the body. It must be due to some sort of shearing becauseof the relative motions but what? And when the table is tilted, the cube does notmove. This would have meant, before Newton, the absence of any force. After him,it meant the forces were in balance. But how?

In equilibrium, the reaction of a surface to the perpendicular component ofweight is the normal force. For hydrostatic equilibrium, this is the same as for ahorizontal table. But for the inclined plane, the motion is impeded by a componentof the weight parallel to the surface while the normal component remains inbalance as a constraint on the motion. The parallel component is the accelerationof gravity, which comes from outside, reduced by the friction. With such immensepractical importance this mechanical effect provoked many studies. The mostsystematic was performed by Charles Augustin Coulomb (1736–1806) who, atthe time, was engaged in various engineering studies stemming from his militaryduties. Coulomb’s law states that the frictional component, whether moving orstatic, depends only on the normal component of the mass and is independent ofits velocity or area. It is therefore a constant fraction of the normal component ofthe force whose value depends only on the properties of the surface with whichthe mass is in contact. For an inclined plane, a block will remain static until


the component of the weight parallel to the plane exceeds the friction. There is,consequently, a critical angle that is independent of the mass given by the angleof repose, W sin θc = µW cos θc from which the coefficient µ can be determined.Several theoretical models were proposed, even incorporating various ideas aboutthe distance dependence of a force. When in contact, the irregularities of thesurface suffice to explain why a body cannot move freely and what can hold it inplace.

Frictional forces have a special significance. Unlike any that can be obtainedfrom a field, they are not conservative. That is, they cannot be obtained from apotential function (see the next chapter). Since the moving body eventually comesto rest, it would seem a natural consequence that such forces are dissipative. This isa fundamental limit to the efficiency of any real mechanical device. But this wasn’tat all obvious at first and remained obscure until work by John Lesley (1766–1832)and Humphrey Davy (1778–1829). And because friction is a damping action, aloss of vis viva, it must be a different kind of force than gravitation. Hermanvon Helmholtz (1821–1894) realized how to include frictional effects in a moreextended framework. His fundamental paper of 1847 was entitled as On theConservation of Forces but in it, we find a more complete principle, conservationof energy, that replaces vis viva and its distinction from vis mortis. Frictionaldissipation was essential for this generalization. Motion resulting from a potentialcan be lost but its “quantity,” the energy, is redistributed between bulk (center ofmass) motion, possible mass losses (for instance, if something breaks), and heat(internal energy). Deformations during collisions, for instance compressions, stressthe body; if this is inelastic some of the kinetic energy is lost but the sum of allpossible forms remains the same. Friction of any kind transforms ordered motioninto something else, heat and possibly changes in the state of the material. Wewill discuss this further in the context of thermodynamics. Force, in Helmholtz’ssense, was not just acceleration butt included action. It also encompassed changesin the state of the body, for instance melting.

Viscosity: The Internal Friction of Fluids

Daniel Bernoulli and Euler had been able to treat the motion of ideal fluids forwhich streamlines maintain constant total pressure and move without frictionrelative to each other, described in a memorable phrase of Richard Feynman as“dry water.” There is a partitioning of the bulk motion that transports the mediumof density ρ in a net direction with speed V and the internal pressure. Thetheorem, due to Bernoulli in his Hydrodynamica (1738), states that in the flowthe sum of the dynamical pressure, P , or the momentum flux, and the internalpressure remains constant along the flow, P + 1

2ρV 2 = constant; here ρ is themass density and V is the mean velocity. This actually defines a streamline.Pressure differences within the medium drive the flow, along with external forcessuch as gravity. These produce continuous changes in speed along the streamlineas well as in time. Euler’s equation of motion describes the hydraulic motionrequiring only boundary conditions to define the specific problem. The pressureacts outward as well as along the flow direction. For instance, in a pipe a change inthe pressure accelerates the fluid but also acts on the walls so the elastic problem


of the expansion of the container becomes coupled to the problem of describingthe motion of the fluid contained.

The difficulty is this is very incomplete. It ignores the drag at the walls becausethe fluid is assumed to be frictionless. Real world observations, for instance canalsand rivers, show that this is too idealized for many applied problems. For example,in channel flow the velocity at the boundaries, perpendicular to the wall, mustvanish and if there is no slip there, so the flow is also stationary at the wall, thena gradient must develop in the pressure that points into the flow. The velocityat the center of the channel is largest, the velocity profile is parabolic relativeto the center of the channel. A nonsymmetric flow, for instance over a flattenedsphere, generates a differential pressure gradient in the fluid and produces lift. Ifyou have a curved surface, the flow accelerates because of the change in directionalong the boundary. We imagine the curved part to be on top. Along a straightedge it is inertial, let’s take that at the bottom. The pressure is therefore differenton the two surfaces and a gradient points upward. This is the principle of theairfoil and the basis of the theoretical treatment of flight. But the Euler equationand Bernoulli’s principle are only the first approximation to the real mechanicsof flight. The treatment has to be extended to include the effects of a distributedfriction, the viscosity, between the moving body and the surrounding medium andwithin the fluid itself. The momentum transfer is to a bulk, deformable mediumthat occurs on many scales depending on the shear.

You may wonder why this long digression in a discussion of forces? For a con-tinuous medium, this viscosity depends on the material properties of the medium.The definition of a fluid is that it is a continuous medium that doesn’t resistshear, or differential distortion. In contrast, from the bending of solids, the Youngmodulus or Hooke constant measure the resistance to deformation. Even for anincompressible medium, the shear is the difference in the momentum between twoparts of the fluid, or between the fluid and any bounding surface. The first solutionfor a fluid was by proposed in 1845 by George G. Stokes (1819–1903), althougha similar result was obtained by the French engineer Navier somewhat earlierusing a very different picture of the interactions between parts of the medium.The importance is that the friction depends on shear, the differential motion ofthe fluid. This dissipation, or “‘loss of head” (impulse), was the essential ingre-dient missing from hydraulic calculations of fluid transport. There are severalsources for viscosity, and these are important for extending our idea of a force.Coulomb’s picture of friction was a contact force between solids. But in a fluid thisarises because of shear and momentum transfer at the microlevel. Macroscopicmomentum transport also occurs because of turbulence, a state of the fluid firstencountered in engineering applications of flows through pipes. Again, the detailswould require far more space than we have available. It is enough to say the forcesinvolves are friction versus the inertia of the fluid, the ratio of which form theReynolds number (a scaling discovered by Osborne Reynolds in the last third ofthe nineteenth century), V L/ν (where ν is the coefficient of kinematic—internalmicroscopic, or kinematic—viscosity and V and L are characteristic scales forvelocity and length within the medium independent of the geometry and boundaryconditions.


ROTATIONAL MOTION AND FRAME OF REFERENCE

After the Principia something was still missing in the mechanical picture: a precisespecification of how to treat space and types of relative motion. It’s almost as ifNewton had “thrown out the baby with the bathwater” by banishing any discussionof centrifugal acceleration from mechanics. For the central field problem withwhich he was most concerned for gravity this was sensible. But it forced a uniquechoice of reference frame, that of a stationary observer who would understand whatmotion is or is not inertial. The relativity of the motion, and its independence offrame of reference, was ignored and including it provided a new understanding ofthe meaning of a force.

I’ll begin this discussion by noting an odd historical fact: Newton never wrotethe iconographic form of the second law of motion in the form that now appearsin every introductory physics textbook, F = ma. As I’d mentioned earlier, thissimple expression was another of Euler’s many contributions to mechanics. It maycome as a surprise that to achieve this compact statement, which is actually notthe same as Newton’s, required almost half a century. But stated this way, forcetakes on a new significance. For Newton, all forces are real, existing things thatrequire sources. For Euler, they are—as the second law states—the agents thatproduce accelerations, for Newton, forces are measured by accelerations. But thetwo are interchangeable and what you call a “real” force depends on your point ofreference. There is a centrifugal acceleration that with a simple change of wordingbecomes a force if you happen to be in a rotating frame and, perhaps, don’t realizeit. Dynamics in a rotating frame appears different, as we saw from the Newtonianbucket problem, because the matter is continually accelerated. Thus, we mustinclude the motion of the “container” as well as motion within the containerwhen describing any accelerations. A simpler form of relative motion was alreadydiscussed by Galileo, rectilinear motion at constant speed. This inertial frame isone that leaves the equations of motion invariant, since the center of mass nowbecomes a reference point. In the equations of motion, we can choose which frameis appropriate, the two should give the same answer for any measurable quantity.We now refer to these as the Eulerian (fixed in a stationary frame outside themotion) and Lagrangian (relative coordinates with respect to the center of mass ormean motion).

To understand this, remember that in a central force problem only the radialcomponent of the motion is accelerated. That’s why for a two body orbit the angularmomentum is strictly constant unless there is a torque, for instance if the centralbody is rotating and extended or deformable by tidal accelerations. In a rotatingframe, any radial motion requires an increase in the angular momentum to stay incorotation so as you move inward or outward from the axis you experience whatseems to be a force that wouldn’t be there if you were in a stationary system. On adisk, this is very apparent, think of trying to walk from the center to the edge ofa carrousel: you feel an apparent deceleration in the angular direction making itvery hard to proceed radially outward. On a rotating sphere, when you are walkingalong a meridian you are actually walking radially relative to the rotation axis.You are moving toward the axis (displacing from the equator) or away (going in


the opposite sense) although you are maintaining a constant distance from thecenter. The same is true when you move vertically (upward motion increases theaxial distance in either hemisphere). Thus the motion, seen in the rotating frame,seems to have a twist—or helicity—and an acceleration always a right anglesto the direction of motion. The sense of this is toward the right in the northernhemisphere of the Earth relative to the direction of motion (and opposite in thesouthern hemisphere). This is the Coriolis force, named for Gaspard Gustave deCoriolis (1792–1843), and it is a dynamical consequence of the kinematics of arotating system.

To be more precise, the force depends on the velocity, not merely the displace-ment. The faster the rate of displacement, the greater the rate of change of theangular momentum, which is an angular acceleration or torque. But the accelera-tion is in the tangential velocity and thus seems to be due to a tangential force. Thepractical example is any motion on the Earth, toward or away from the equator.The deflection depends on the rotational frequency of the surface, the rotationfrequency of the Earth, and the latitude, which determines the angle betweenthe local vertical and the direction of the rotation axis in space. The existenceof this force was dramatically demonstrated in 1851 by J. B. Leon Foucault in(1819–1868) with a very long (11 meters) pendulum suspended from ceiling of themeridian hall of the Paris Observatory and later, in a public demonstration, fromthe dome of the Pantheon in Paris. The Coriolis acceleration also dominates themotions of the atmosphere, especially local systems of high and low pressure andexplained the laws for the directions of winds.11

Thus, relative to an observer at rest—that is, not in the rotating frame—thereis always an angular motion associated with any change in position. This was firstdiscussed by Euler. Even an observer who isn’t moving within the rotating system,for instance someone sitting on a carrousel, is still being carried around at theangular velocity, ω, relative to an external observer, who is also at rest but in a trulystationary frame. This is not invertible. Although we can change the sense of therotation by making the whole universe rotate in the opposite direction from purelykinematic point of view, the essential difference is that rotation, unlike rectilinearmotion, is never inertial: the motion takes place in at least two dimensions thatchange in time and is therefore always accelerated.

This was the basis of Ernst Mach’s (1838–1916) discussion of force. It gets tothe core of what inertia is all about, that there is some sense in which relativemotion is not symmetric. Newtonian cosmology assumed an absolute referenceframe and an absolute time. But somehow the moving system “knows” that it isn’tinertial because forces appear in one frame of reference that another doesn’t detect,seeing instead a different motion. The Coriolis and centrifugal forces in rotatingframes are the principal examples. Mach’s solution was that even if all motionsare relative, on the largest scale there is a semi-stationary distribution of the massof the universe as a whole with respect to which all local motions are referred.This became known, much later, as “Mach’s principle.” It is still an open questionwhether this is the origin of inertia, as we will see when discussing the generaltheory of relativity and gravitation in the twentieth century later in this book.

A way of seeing this uses another device invented Foucault in 1852, thegyroscope. Although it is now a children’s curiosity, it is a profound example


of how non-intuitive the effects are of rotation and angular momentum conserva-tion. Euler showed that a non-symmetric mass will tumble around a principal axisof inertia so its apparent rotation axis seems to shift over the surface of the bodyon a period that depends on the distortion of the body from a spherical shape andon the rotation period. This freebody precession, in which the angular momentumremains fixed in space but the instantaneous axis of rotation shifts around, wasfinally detected for the Earth by Seth Chandler (1846–1913) using the motion ofthe north celestial pole. Remember, an observation by an observer on the surfaceis made in a rotating frame. The period is about 14 months for a distortion of about0.3% and a rotation period of one day with an effective displacement of the orderof meters. The precession of the equinoxes is a much greater effect, caused bythe distortion and the gravitational attraction of the Sun and Moon on the tipped,spinning Earth. Foucault’s gyroscope consisted of heavy disk that spun rapidlyaround an axis that was attached to an outer support ring that could be either sus-pended on a cord or balanced on a fulcrum, much as it is nowadays. When tiltedthe disk is torqued like a lever and tends to right itself. But in so doing the angularmomentum must change direction which produces a motion around the vertical,the direction of the gravitational acceleration. The disk is far more flattened thanthe Earth but the principle is the same. Foucault realized that the device providesa reference for the direction of gravitation and it was soon employed as a stabilizerfor ships and, eventually, for satellites and missiles.

NOTES

1. The texts are taken from the translation by Cohen and Whitman (Newton 1999).This is the best introduction in the English language to the history, context, and contentsof Newton’s monumental work and I urge you to read it.

2. In effect, he showed that Galileo’s treatment of freefall is the limiting case whenthe distance traversed is small with respect to that from the center of the force so theacceleration is approximately constant (see the “Universality” section).

3. This is the line passing through the foci of the ellipse that connects the points ofclosest and farthest approach to the central mass.

4. In many textbooks in introductory physics it’s popular to say the centrifugal accel-eration is fictitious. From the viewpoint of an external observer that’s certainly true andthat’s what Newton understood. But when you are reckoning the accelerations, if you’renot moving inertially there is a force. It doesn’t do to simply dismiss it, and this was oneof the critical steps that led to the general theory of relativity and that we will meet inEinstein’s principle of equivalence in a different form.

5. This is how Charles Darwin summed up his On the Origin of Species. It can equallybe applied to Newton’s magnum opus and perhaps in the same guise. Darwin used a similarprinciple to that employed in the Principia, explaining the phenomena of diversity of lifein terms of universal mechanisms, the origin(s) of which were yet to be discovered.

6. In fact, this is still true. Although the pendulum has been replaced by bars andbalanced oscillators driven at specific reference frequencies in modern gravitometers, theprinciple is essentially unchanged. A loaded pendulum, normal or inverted, stable orunstable, is used to measure the acceleration of gravity. An even more dramatic exampleis the arrangement of the gravitational wave interferometer Virgo that uses an invertedpendulum to serve as the test mass for detecting change in the path of the light.


7. I’m using here the Motte translation of 1727, prepared under Newton’s guidance(that is, direction), since it is the one version that immediately connects with the intent ofthe Principia and was also read by his contemporaries who were less schooled in Latin.

8. Halley was the first to see the contradiction in this and although it departs fromthe main thread of our discussion of force, it is instructive to see how rich the Newtonianworldview was in new problems. If, Halley argued, the light from distant bodies diminishessimilarly to their gravitational influence, then the it would seem that an infinite universecontradicts experience. Because the number of sources intercepted by a cone of fixed solidangle grows as r 3 while the intensity of any individual star diminishes as 1/r 2, the summedintensity should increase linearly with increasing distance. Hence, the night sky should beinfinitely bright, an unacceptable conclusion. Yet to imagine a world with an edge wouldhave two equally unacceptable consequences. There would have to be a boundary, at finitedistance, beyond which there would be only void. And the stability of the cosmos againbecomes a problem. See Jaki (1973), Harrison (2000), and references therein for historyand ultimate the resolution of what became known as Olber’s paradox.

9. This was not the first “new planet,” of course, Uranus had been discovered by chanceby William Herschel in 1781. But there were other, so-called minor bodies, that had beendiscovered before Neptune was predicted. The first was Ceres, the largest of the asteroids,followed by Pallas and Vesta at the start of the nineteenth century. Their number steadilyincreased over the next two centuries.

10. This technique has been extended in the past few decades to other visualizationtools, especially interference of lasers reflected from the surfaces, and even dynamicalimaging is possible. But the basic idea has not changed since Chladni’s first investigations.

11. A low pressure region produces locally radial motion as air moves from the sur-roundings into the low. This produces a vortical motion that is counterclockwise, orcyclonic, in the Northern hemisphere. It isn’t true, by the way, that this is observable inbathtubs and sinks or even tornados; the helicity of these flow is imposed at their source,pipes and downdrafts in wind-sheared cold atmospheric layers, and is due to the conser-vation of circulation rather than the large scale Coriolis effect that dominates structuressuch as hurricanes.

5

FORCES AND FIELDS

All physicists agree that the problem of physics consists in tracing the phenomenaof nature back to the simple laws of mechanics.

—Heinrich Hertz, The Principles of Mechanics

Even while the Principia was being read, digested, and debated, a complementarypicture to Newton’s was taking shape on the continent at the end of the seventeenthcentury where the Cartesian influence still held sway. The nagging difficulty wasthat there was an essential ingredient missing in the Newtonian program: whilethe effect of gravity could be described, both its source and transmission remainedunexplained. Notwithstanding his metaphysical justifications for this lacuna, tosay this bothered Newton’s contemporaries is an understatement. It was the majorstumbling block, in the context of the Mechanical Philosophy, that an action shouldbe invoked without cause. Newton’s own analogies served poorly to assuage themisgivings of his readers, especially Leibniz and the continental mathematicians.It seemed to call for some sort of action at a distance, an effect that almostmiraculously issues from one body and affect another regardless of where it islocated. To this point, all treatments of force had required some sort of contact.The Cartesian vortex, acting through a subtle fluid that pervades space, at leastsatisfied this requirement. But Newton had argued, in the extended polemic of thesecond book of Principia, that this would not suffice to produce the inverse squarelaw. While this may seem to be the heady stuff of philosophical debate it wasfar more. It called for a fundamental change in how to reason physically aboutNature.

The idea of a“field” as the origin of motive power was born in a sort of theoreticalcompromise and even with the antipathy to “attractions” and “influences” aswords, the concept of field was very vague at first and was couched in languagethat resembled the Cartesian plenum. But it was, somehow, different. Instead ofhaving a dynamical role in the creation of gravitational attraction, the field was


something affected by masses to produce the desired effects. Spatial differencesin the intensity of the field became the forces and a body was assumed to movecontinuously under their influence because the medium—whatever it might be—is continuous. This permitted the source of the action to reside in the massivebodies which, you’ll recall, had been defined rather than deduced in Newton’sprogram. Their influence on other masses, their attractions (the very conceptthat had been so philosophically perplexing), became the continuous changesin the density of this universal medium. An interesting symbiosis developed.Increasing familiarity with the properties of fluid and elastic substances produceduseful analogies, and the foundational issues spurred further work along theselines.

This shift is vitally important for understanding all subsequent developments ofphysics. Dynamical questions were now to be posed in two steps. The first requiredpostulating what kind of field was involved given its source (although still not its“nature”) and then deriving the force through the its spatial dependences. Then,knowing how the force behaves, to compute the trajectory of a mass (or a collectionof masses) thus acted upon. The basic problem was thus buried, not solved, andwith this new-found freedom it was possible to make rapid progress. In effect, toknow—or hypothesize—the field was to know the force. Or, conversely, to knowthe dynamics was to know the force and through the motion to determine the field.In this view, the physical scale at the phenomena occur doesn’t matter. If forcesare universal, although perhaps different in nature depending on the agent andcircumstances, the same methodology can be applied.

THE ATOMIC CLUES

You’ll recall that Galileo and his followers had been concerned with the nature ofair and the vacuum. Experiments were possible after Robert Boyle’s (1627–1691)invention of the air pump. This provided a means for artificially changing thepressure in sealed volumes under controlled conditions. With it, Boyle discoveredthe first law for the internal pressure of gases, that the reaction force increaseson compression. You’ll recall that liquid water, being an incompressible medium,produces this reaction depending only on its weight above any level. This propertywas essential in early arguments about continuous media. A gas can change itsdensity so its weight changes the pressure also by compression. The atomisticpicture was perfectly suited to explain this behavior, as Daniel Bernoulli didby assuming a gas to be composed of particles that are in constant motion andcollide both among themselves and with the walls of the container. The argumentis beautifully simple. If we assume all particles have an momentum (or impulse)p, and collide elastically, then when they hit a wall they bounce without changingthe magnitude of this momentum. The kick, �p, given to the wall is then oforder 2p since they simply reverse direction (remember the third law of motion).There is another factor to consider, the mean angle of incidence, because theparticles are assumed to arrive from any direction and thus only a small fractiondeliver the kick along the surface normal. The number of such collisions in aninterval of time depends on the rate of arrival of these particles at the wall. If Nis the total number in the container of volume V of velocity v and mass m, the

Forces and Fields 101

rate of arrival of the particles at the wall per unit area per unit time is Nv/V .Since the force is the time rate of change of the momentum, mv, in an intervalof time is N�p/�t , and the pressure is the force per unit area, this gives thepressure law P V = constant. This also provided a scaling law for the internalpressure, P = p2 N/(mV ) for a gas of identical particles. Furthermore, becausethe impulse—the kick on collision—is delivered perpendicular to the walls of thecontainer this explained Pascal’s law for pressure. The result provided at once asuccessful explanation for how to apply the new force concepts to materials andalso strengthened the atomic viewpoint already implicitly contained in Newton’sconception of both matter and light.

Newton had, in fact, published a possible extension of his conception of force tofundamental interactions and the atomic nature of matter but not in the Principia.That remained a work addressed to the picture on the large scale. However, heallowed himself more speculative freedom in his other work, the Opticks. Unlikehis magnum opus, this was first written in English. The first edition was publishedin 1704. It summarized Newton’s studies of light and optics but also included asmall set of questions (queries) in a closing group of essays on the nature of light,its propagation, and the relation of optical phenomena to those he had developedin the Principia. The work quickly appeared in a Latin edition prepared by SamuelClarke with Newton’s supervision and containing a revised and augmented set ofqueries. With the second English edition in 1717 we have Newton’s statement ofhis “hypotheses.”

Recall Newton’s absorbing interest in alchemy. For the final query, number 31in the last edition, he pulled out all the stops, composing what can be read as asymphony. The text opens with the direct statement of the theme:

Qu. 31: Have not the small particles of bodies certain powers, virtues, or forces, bywhich they act at a distance, not only upon the rays of light for reflecting, refracting,and inflecting them, but also upon one another for producing a great part of thephenomena of Nature? For it’s well known that bodies act upon one another bythe attraction of gravity, magnetism, and electricity; and these instances shew thetenor and course of Nature, and make it not improbable but that there may bemore attractive powers than these.

There follows a long catalog of chemical phenomena that show how combinationsrequire some sort of interaction that, unlike gravity, depends on the specificproperties of matter. He then opens the second movement, inferring by inversereasoning the probable properties of the hypothesized forces:

The parts of all homogeneal hard bodies which fully touch one another stick togethervery strongly. And for explaining how this may be, some have invented hookedatoms, which is begging the question; and others tell us that bodies are gluedtogether by rest (that is, by an occult quality, or rather by nothing); and others,that they stick together by conspiring motions (that is, by relative rest amongstthemselves). I had rather infer from their cohesion that their particles attract oneanother by some force, which in immediate contact is exceedingly strong, at smalldistances performs the chemical operations above mentioned, and reaches not farfrom the particles with any sensible effect.


Returning again to specifics, he describes how the forces might build macroscopicbodies for which, in the end, gravity alone dominates. And then he generalize theinteractions:

And thus Nature will be very conformable to herself and very simple, performingall the great motions of the heavenly bodies by the attraction of gravity whichintercedes these bodies, and almost all the small ones of their particles by someother attracting and repelling powers which intercede the particles. The vis inertiaeis a passive principle by which bodies persist in their motion or rest, and resistas much as they are resisted. By this principle alone there would be no motion inthe world. Some other principle is necessary for putting bodies into motion; andnow that they are in motion, some other principle for conserving the motion.

Newton then discusses collisions, showing that by composition of momenta theremay not be completely in agreement with the principle of conservation of momen-tum. This leads, at last, to a picture of atoms as fundamental units of which bodiesare composed, ending with the affirmation that “therefore I scruple not to proposethe principles of motion above mentioned, they being very general extent, andleave their causes to be found out.” The text then ends with a deistic coda and thepiece draws to a close, the final chords being the General Scholium.

FIELD AS FORCE

For Newton, forces remained impulsive, even in the limit of an almost infinitenumber of collisions. Something emanates from a mass that produces an influenceon distant bodies and changes the motion of neighboring masses accordingly.Ignoring the problem of propagation, this construct sufficed to empty the worldof its ether and vortices. Voltaire reported this in his English Letters noting that,in France, one finds a world full of fluids and vortices while in England it wasempty. That’s why modifying the Cartesian program was not so difficult once theNewtonian principles were included. It wasn’t necessary to completely abandonthe fluid as an explanation as long as it acted according to the laws of motionand obeyed their constraints. The advantage was that the source for gravitationmight still be elusive, but its action could be more simply explained. We rename it,and call it a field, and imagine it as a continuous substance. Within the mechanicalphilosophy this required that spatial changes of this field produce the force.Conservative forces such as gravity, the gradients are always in the same directionas the displacements so the increase in motion comes from the work done toplace the body in its starting position. In order to produce a change of state ofmotion, this something must somehow be unequally distributed among things inproportion to their mass. If this seems too vague for your taste, it is nonethelessremarkable that even without specifying what this field is, considerable progresswas achieved by concentrating only on how it acts. Just as pressure causes motionwhen it unbalanced, so it may be with this field. Spatial gradients produce a netforce that is only counteracted—the reaction—by the inertia of the body.

It seemed possible, within this conception, to take up the challenge of theQueries but few tried beyond lip service and change in vocabulary, when theyweren’t lamenting the “occult qualities” that seemed to be again entering physical


thought. One who did was the Jesuit polymath Roger Boscovich (1711–1787).He assumed static forces that depend only on the proximity of atoms but thatthe fields by which they interact may have much shorter range than the inversesquare law. “It will be found that everything depends on the composition of theforces with which these particles of matter act upon one another: and from theseforces, as a matter of fact, all phenomena of Nature take their origin” (TheoriaPhilosophiae Naturalis 1758). An important hypothesis was the existence of anintrinsically repulsive core to the atom, one that prevented collapse of matter onclose approach. This catastrophic state is unavoidable in Newton’s cosmology fora gravitation-only universe. Even while not assigning a cause to this atomic levelforce, Boscovich could not only explain the different states of matter but alsoprovided a framework for understanding chemical combinations. He consideredthe field to be the fundamental property of matter, whose spatial dependencecould be very complicated, having alternating zones of attraction and repulsion,postulating a universal field whose particulars depend of the “type” of atom. Herehe made a significant change with respect to Hellenistic and Roman atomism. Hisatoms are really points endowed with mass (inertia) and the field. Electrical andmagnetic phenomena could have furnished close analogs but instead Boscovichconcentrated on the form that would be required for the law. He also realized thatthere could be “trapped states” if the field reverses sign, points of equilibriumaround which the body could oscillate, but did not explore this further and itremained only implicit in his construction.

GEORGE GREEN: THE ORIGIN OF POTENTIALS

Even with the investigations of Laplace, Legendre, Lagrange, and Poisson, bythe second decade of the eighteenth century the concept of field remained rathervague. The analytic link was still lacking. That was provided by the Englishmathematician George Green (1793–1841). He remains a shadowy figure in thehistory of physics. Largely self-educated, far from the centers of the scientificactivity, with a privately published single memoir he introduced the methods andframework for uniting the Newtonian and continental programs that, when finallyfully appreciated, revolutionized the concept of force and changed the course ofphysics. Green didn’t simply introduced the term and concept of the potentialin his 1827 memoir An Essay on the Application of Mathematical analysis tothe Theories of Electricity and Magnetism. He showed that any force, electricalor gravitational, can be found from the spatial variation of a field of potential,a function that depends only of position without any particular directionality inspace.1

The force law for gravity was originally derived by Newton from orbital motionby assuming that locally—within the orbit—the centripetal force constrained themotion. The first derivation of a general mathematical expression for how to unitethe field and its sources with the force it produces was achieved by Pierre-Simonde Laplace (1749–1827) at the end of the eighteenth century. He assumed a masswithin a space that is the source of the field which must give only a net radialforce. What lies outside of this mass was unimportant as long as its influence couldextend to infinity. The shape of the body didn’t need to be spherical, Newton had


normal

equipotentialsurfaces

Figure 5.1: The construction of gradients through equipotential surfaces, anillustration of Green’s idea of the potential.

already demonstrated that such a case is trivial in the sense that the resultingfield is strictly radial. Laplace allowed for an arbitrary shape but remained outsidethe body so the force must, asymptotically with increasing distance, approachspherically symmetry. In this case, there can be no net divergence of the force: ifwe draw a sphere, or a spheroid, around the source of the field and extend its radiusto infinity, the force must asymptotically vanish. The equation describing this wasthe starting point for the generalized concept of a field of force, stating that thereis no net divergence over all space of the acceleration produced by a mass. It wasstraightforward, then, to include the effects of extended mass distributions. Poissonaugmented the Laplace equation to treat a medium with sources, an extended massdistribution, was a relatively small alteration. Again, the boundary condition isthat the mass is finite, if extended to infinity the force depends only on the amountinternal mass. Each element of the mass attracts the test body so the net force isreally a sum over all masses that are anywhere other than where we happen tobe standing. Said differently, it is a convolution—literally, a folding together—ofthe density distribution in the sense that the force at a point within some densitydistribution is produced by the sum of all influences outside that point.

Now let me expand on this point for a moment. Laplace extended the forceconcept by introducing a field equation, the representation of the field as a con-tinuous property of the space surrounding a mass. If we are outside all masses,then their influence, their collective force, has a very specific property. This is anessential feature of action at a distance, that the field once established fills spaceonly depending on the distance between the affected bodies. Imagine a sphericalmass in isolation, or at least far enough away from other bodies for their effect tobe negligible. The force is then radial, directed toward the center of mass. Thesource of the field can then be treated as a point and we take the rate of divergenceof the influence through successively larger spheres circumscribing the source.Saying that this divergence vanishes is the same as saying that whatever “effect”passes through a sphere of radius r will pass through a sphere of radius r ′ > rand also r ′ < r. The same is true for light, the flux from a constant source is con-served so the amount of energy passing through any element of a circumscribed


surface decreases as the inverse square of the distance; the total amount of lightpassing through the whole surface remains constant. Newton had hinted at thisanalogy in the Opticks, Laplace showed that the equation for the field then givesa general solution for any coordinate system and permits the calculation of theforce, through an auxiliary function, by saying that if the force is the gradient ofsome functional representation—the field—then the divergence of that gradientvanishes. As I’ve mentioned, Simeon-Denis Poisson (1781–1840) extended this tothe interior of a distribution of arbitrary density, demonstrating that the divergenceof the force depends on the spatial distribution of the density of matter. For a pointsource this reduces to the Laplace equation. For electrical media, Gauss derivedthe same equation using charge instead of mass, that the electric field passingthrough any completely encompassing surface is normal to the surface and is onlydue to the charge contained herein. The mathematical generalization of this resultto any arbitrary scalar quantity is now called Gauss’ theorem.2 We know fromthe introductory remarks to the Essay that Green had read Laplace’s monumentalMecanique Celeste —at least some of the formal parts dealing with the represen-tation of functions in spherical coordinates, Fourier’s Theorie analytiques del lachaleur, the relevant papers by Poisson (but perhaps only in translation in theQuarterly Journal of Science), Biot’s book Traite de physique from 1824—but wasessentially self-taught and isolated from the centers of English and Continentalmathematical physics. The problem Green was examining, how to find from theelectric field the distribution of the internal charges, had already been confrontednearly a half century before in a completely different setting, that of gravity. Greendefined the potential as a function, a scalar whose gradients produce forces butwhich is not itself measurable. His is the first use of the term although the conceptwas already built into vis viva. This step might be described as the bridge betweendead and live forces, the field comes from the properties of distributed matter butthe action of the distribution produces motion.

This potential function wasn’t merely a technical modification of mechanics. Itprovided a way to explain and visualize forces. Now it was possible to use inertiaagain geometrically, not only for motion but also for statics. One could pictureof field without asking about action at a distance, without emission or absorptionof the agent responsible for the forces. It meant only imagining a landscape inwhich, for instance, a body placed at some point would be stable or unstablearound its equilibrium. Instead of thinking of lines—trajectories—and directionsof influences, it allows you to imagine surfaces and gradients. The mathematicalconnection with geometry, and in particular with differential geometry, was adecisive step. It’s possible to say, for instance, that the potential is the field,that the forces are therefore produced by changes in the landscape in space and,eventually, in spacetime. For instance, if there is little change in the field at anyplace, or if you take a region small enough (or far enough away from a central sourcefor the field, the local change in the direction of the surface normal gradients is assmall as you want and therefore the force appears both vertical and the potentialsurfaces flat. How like the problem of mapping in geography. If you take a regionsufficiently small in radius, the world is flat. As the view extends farther, towardthe horizon, the surface may appear curved. That is, if the view is clear enoughand that is always true in our case.3


Geodesy and the Measurement of the Density of the Earth

But to understand Green’s revolution, we should step back for a moment. Geodesywas the child of the eighteenth century, a part of the systematic mapping of theEarth’s surface. If the previous two centuries were expansionist, the 1700s weredriven by a need for precision to maintain the products of that expansion. Europeancolonies were spread across the globe and the seas were the open thoroughfare topass between them efficiently. But it was necessary, on land and by sea, to haveaccurate maps. This required, in turn, precise knowledge of latitude and longitudeand of the shape of the planet. For instance, in siting a meridian line—even withan accurate clock—it is necessary to know where the vertical is and also thealtitude relative to a fixed reference frame. This was provided by gravitation. Amodel for the Earth was also required for altitude measurements since the swingof a standard pendulum (a “seconds pendulum”) could provide the reference fortime but required knowing the distance from the center of the planet. During theFrench expedition to Peru, under the auspices of the Academie des Sciences,to measure the length of the meridian arc for geodetic reference, Pierre Bouguer(1698–1758) developed one of the standard corrections for terrain gravity. He tookaccount of both what is now called the “free air” correction, which is an adjust-ment for the altitude of the observing station relative to the reference spheroid,and for the mass of the underlying material, now called the “Bouguer correc-tion.” In effect, these are corrections not to the acceleration but to the gradient

Figure 5.2: In the background of this portrait ofNevil Maskelyne, notice the mountain that wasthe object of his study of the gravitational at-traction of the Earth. Image copyright History ofScience Collections, University of Oklahoma Li-braries.

of the acceleration and require a knowledge ofthe geology of the underlying material or at leastan estimate of its mean density.

A fundamental test of Newtonian gravitation,albeit an indirect one outside the controlledspace of the laboratory, was Nevil Maskelyne’s(1732–1811) measurement of the deviation of aplumb line by the presence of a nearby moun-tain. It became one of the first geophysical mea-surements, the mass of the mountain was deter-mined by the deviation and this was then, alongwith geodetic sitings of the structure, to obtainthe contrast between the density of the surfaceand that of the bulk of the Earth. The result, thatthe density contrast was nearly a factor of twobetween the superficial and deep matter, mightseem a mere curiosity. Yet this was a startingpoint for modeling a part of the Earth inacces-sible to direct measurement and was importantnot only for the initial technical problem buteven had cosmic significance. With this deter-mination of the density profile, albeit only twoextreme values, it was clear that the planet ismore centrally condensed than the mean esti-mate based on the acceleration of gravity and


the radius of the planet. This, in turn, affected the calculation of the lunar orbitand of the tidal effect. It was the beginning, as well, of the investigation that cul-minated in Henry Cavendish’s (1731–1810) measurement of the density contrastof the Earth using the laboratory determination of the proportionality constant forgravitation, the universal value of which still lacked precision at the end of thecentury. The Cavendish experiment of 1798, as it is now universally called, used atorsion balance, a technique perfected for electrostatic measurements by Coulombin 1784. The principle is brilliantly simple, even more so for masses than chargesbecause the relative magnitude of two masses was far more easily determinedthan the relative quantity of electrification. Two test masses were attached to asuspended beam which was maintained in a thermally and pressure controlledcase. A thin wire was attached to a micrometer and two masses, suspended froma second arm, were swung from rest to a fixed angular displacement. The testmasses, thus accelerated, displaced and oscillated for hours, and the deflectionwas measured by a small mirror mounted on the wire to reflect a light beam to awall where the deflection could be more easily observed and recorded. AlthoughCavendish intended this to be a measurement of the density of the Earth, its mainresult was the precise value for G, the strength of the gravitational attraction, sothe proportional statement of the gravitational force law between two masses, M1

and M2 separated by a distance r could be replaced by a precise dimensionedstatement:

F = −GM1 M2

r 2

The measurement was repeated, with increasing accuracy, during the 1800s, sum-marized by C. V. Boys (who asserted that the principal result of the Cavendishexperiment was the value of the gravitational constant), and more refined mea-surements, now using different techniques, continue through this century.

The Potential Function

Now, after this interlude, we return to George Green. The Essay was greeted witha deafening silence, at least at first. Few copies were printed, and surely manythat were receive by even the subscribers went unread. Green’s time in Cambridgewas longer as a student than as a Fellow, a post he held for less than two yearsuntil he returned to Nottingham in 1841 where he died (the cause is unknown) amatter of months later. Copies of the work were also sent to such noted scientistsas John Herschel (who had been one of the subscribers) and Jacobi, but it isdoubtful they—or anyone else—initially appreciated the extent of the change ofmethod and power of the mathematics contained in the work. How it came to beknown is as famous as it is instructive to the student to always read the literature. In1845, a fellow Cambridge student of mathematics and natural philosophy, WilliamThomson (later Lord Kelvin), was preparing to leave for France for an extendedperiod of study in Paris when he read a notice in Robert Murphy’s 1833 paper oninversion methods for definite integrals in the Cambridge Philosophical Societyproceedings regarding Green’s theorems. He received copies of the Essay fromhis tutor at Cambridge, William Hopkins (it is strange that despite Green’s high


ranking in the mathematical Tripos exam and his sojourn at the university he wasalmost forgotten by the time of this event). It changed his approach to physics andhaving found it, at last, Kelvin became the principal exponent of Green’s ideas inEngland and Europe.

The striking feature of the potential function is that many problems becomemuch clearer with this new mathematical representation of the force. In the Prin-cipia, Newton had shown that the attraction of a mass depends only on its geometryand not on its internal distribution. More important was his demonstration thatfor a homogeneous symmetric body, the gravitational attraction is due only to themass located interior to the bounding surface. In spheres, for instance, this meansthe equilibrium of a body depends only on the mass located within any radius.This can be generalized for ellipsoids, a step first taken by Maclaurin. For in-stance, the shape of an incompressible body, a liquid, that is self-gravitating andin hydrostatic balance will conform to surface of equal potential. In this case, theboundary shape of the body, and its interior structure, are completely determined;the equilibrium surface is one of constant pressure (recall the implicit use of thisprinciple by Archimedes). Thus, in the language now familiar from Green, thepressure is constant on equipotential.

There were many problems that could not be simply overcome. For instance,the question of whether the representation of the surface is unique was not clear.While it seemed that a distribution of sources would produce a unique force at anypoint in space it was not clear that the distribution could be found from the mea-surement of the forces themselves. The mathematical problem, that of boundaryconditions and how they should be specified to solve the problem of the structureof the field, became a central area of research throughout the nineteenth andtwentieth centuries. But using this concept of potentials did enlarge the range ofquestions that could be addressed in celestial mechanics. For example, the tidalforces felt by the Earth from the Sun and Moon had required complex formalismto resolve the forces in space and time. Now, with a single function representingthe equipotentials the problem could be reduced to finding a single scalar func-tion of position that, in an orbiting reference frame, could then depend on time.The oceans reside along an equipotential surface, precisely as do the charges inGreen’s idealized conductor, their surface being determined by hydrostatic bal-ance. Therefore it is essentially a problem of geometry and analysis to computethe shape of the body and the forces that result on any mass within or on it. In thiswe see the beginnings of a much broader link between fields and force, the originof inertia.

Why only gravity? There are other forces, so why was/is it so hard to makesense of this particular force? We return to the fundamental Newtonian insight:gravitation is a universal property of mass. The inertia is an attribute of a massin motion or its resistance to acceleration, and therefore anything that changesthe state of motion should have the same status. It is a force like all forces. Thespecial place of gravitation is that it forms the background in which everythingelse occurs. But there are many such forces, not only gravity. In a rotating frame,the Coriolis acceleration depends on both the rotation frequency of the frame ofreference and the particle velocity.

Now that the link between the gravitational force and the potential was clear thenext step was to apply it to the structure of cosmic bodies. The problem had been


examined before Green, by Newton in Book I of Principia where he demonstratedthe necessarily oblate form of the Earth, and by Maclaurin. Green also lookedat this problem. In all cases the studies used liquid masses, incompressiblebodies in hydrostatic balance, so the form corresponded to the equipotentialsurfaces. The new wrinkle was a purely nineteenth-century conception. Havingfound that the bodies are in equilibrium under a specific range of conditions, forinstance for certain rotation frequencies, the search began to understand if thesebodies are stable. Since the time dependence of gravitational interaction couldbe either resonant or not for orbits, it was interesting to examine whether thesedifferent self- gravitating masses are stable. As the question of the origin of thesolar system became more relevant, as the timescales became clearer (this waspartly from geological reasoning and partly from thermodynamics), as it becameclear that the system needed to have a mechanical explanation for its origins,mathematicians and physicists looked with increasing interest to the problemthat became known as “cosmogony.” The problem persisted. Nearly two centuriesafter Green, the study of the stability of self-gravitating bodies remained—and stillis—unexhausted.

THE PRINCIPLE OF LEAST ACTION

The prehistory of the least action idea begins with Pierre Fermat in the 1660swho, in treating the passage of light through a medium, obtained Snel’s law forrefraction as a consequence of the principle of least time. At an interface betweentwo optical media of different densities, if we take the length of a chord to be thesine of the angle, θ , the law of refraction states that the ratio of the chords onopposite sides of the interface, sin θ , is constant regardless of the incident angleand depends only on the change in medium across any interface, even a virtualone (for example, in the case of a continuously changing density). This can bestated as sin θ ′ = n sin θ where n is called the “index of refraction.” Now we takea medium of variable index of refraction and imagine that for any depth we cantake some small distance over which n is constant. Then if we assume that to adepth in the medium the time to pass is given by the speed of light divided by thetotal length of the path, we can ask what is the minimum time to traverse a path or,conversely, what path will give the minimum time of passage or least action of thelight? This is a variational problem, examining all possible paths irrespective oftheir curvature, to find the shortest total course for the light for a specified choiceof the index of refraction. Fermat, in developing this variation principle, showedthat the solution for constant index of refraction is a straight line and otherwiseis a curve whose curvature depends on the depth dependence of n. You’ll noticethis requires knowing how to take infinitesimal steps to construct a continuoustrajectory for a particle—including light—that required the calculus that was soondeveloped by Newton and Leibniz.

By the middle of the eighteenth century, physics had begun to spawn manyinteresting mathematical problems and methods, and this problem of least actionwas one of the most intriguing. Jean le Rond d’Alembert (1717–1784), following aproposal of Maupertius in the 1740s, made it the central principle for his derivationof mechanics. Extending Fermat’s principle, he postulated that “among all paths aforce can produce, the one followed by a particle under continuous action of a force


Figure 5.3: Joseph Louis Lagrange. Image copyright Historyof Science Collections, University of Oklahoma Libraries.

will be the one requiring least work.”This is the one for which the total ac-tion, which is related to the vis viva ofthe body, has the minimum change intime. But how to define action was notso simple. It fell to Leonard Euler andJoseph Louis Lagrange (1736–1813) inthe 1750s to develop this idea to itscurrent form. It was later expanded byLagrange and became the foundation ofhis analytical treatment of force prob-lems in his Mecanique Analytique, firstpublished in 1788.4

For Euler and Lagrange, this ac-tion principle became the foundation ofmechanics and, therefore, a surrogateto force. Instead of thinking of actionand reaction, the force is assumed tobe derived from a continuous potentialfunction whose spatial gradients deter-mine its components. In two memoirs in1760, and in subsequent editions of theMecanique Analytique) Lagrange intro-duced a generalized variational princi-ple based on action to obtain the equa-tions of motion. We have seen withLeibniz that any particle with a momen-

tum p will move in time with a vis viva whose sum over time is the action. Instead, ifwe start with a new definition for the action as a sum over infinitesimal changes inthe kinetic energy during some interval of time. Lagrange assumed this quantity—what is now called the Lagrangian, the difference between the vis viva and thework—would be minimized among all possible choices of arbitrary displacements.Leibniz’s definition of action, the product of the vis viva with the interval of timeduring which the motion takes place, is not the same as that adopted by Lagrangeand Euler. The potential, that is the external force, doesn’t appear and thereforethis is only a conservation law for a free particle that is only valid for inertial mo-tion. Instead, Lagrange and Euler explicitly included the effects of a backgroundforce. Since the work is the force multiplied by the displacement, the variationalproblem now becomes finding the way the kinetic energy changes such that thetotal variation in the work plus the kinetic energy vanishes. A rigidly rotatingbody also exhibits a “constrained” motion, its moment of inertia remains constant.For any mass, or system of masses, characterized by such constraints, Lagrangefound they can be added to the variational problem because they remain constant.Take, for example, a system that maintains both constant energy and angularmomentum. Then since each will have vanishing variation, the sum of the varia-tions must also vanish. Although this formalism introduces nothing fundamentallynew to mechanics, it clarified the link between force, in these cases weight, and


vis viva (kinetic energy) through the applications of D’Almbert’s principle and leastaction. But when introduced it provided substantial insight and, most important,heralded the application of small perturbations to generalized dynamical problemsfrom celestial mechanics, thermodynamics, and eventually quantum mechanics.It is a change of viewpoint. To explain this further, consider that the problem ofmotion, not its stability, dominated the initial development of classical mechanicsafter the Principia for nearly a century. A vanishing net force is enough to insureinertial motion but it cannot provide the answer to the question “is that statestable?” In an increasingly broad range of investigations this became the focus. Inparticular, when in celestial mechanics persistent (secular) changes were obtainedfor the planetary orbits on relatively short (even by biblical standards, notably inthe orbits of Jupiter and Saturn) it was imperative to determine if these were trulymonotonic or periodic. The definitive answer for the Jupiter–Saturn system wasprovided by Laplace from a perturbative treatment: the changes are periodic buton very long timescales and the system is secularly stable.

For fluids the problem is more difficult precisely because they’re not rigid. Butremember Bernoulli’s conception of an ideal fluid as a collection of streamlines.Each carries a mass density and has a volume as well as a mean velocity. Theyhave, consequently, a momentum density and an angular momentum density. Ifwe constrain them to not overlap, these streamlines can also form vortices andthe motion can be treated analogously to the rigid rotation case. Instead of abody following a trajectory, the fluid is itself the trajectory since it is space-fillingand the laws that govern the motion of single particles can be extended to acontinuum. This was begun with Euler in the 1750s and extended through thefollowing century as the concept of, at first an incompressible continuous mediumand then a compressible one, developed. Here the problem of stability was alsomore difficult but the idea of using energy to derive forces instead of the other wayaround, to use the work performed by or on a system instead of asking questionsabout the external forces, gradually took center stage. For stability, a body mustachieve its minimum energy and it must do so without changing its structure. Forrotating fluid masses, for instance, this produced phenomena not encountered insingle particle cases.

Hamilton and Action

The variational method was further exploited by Karl Gustav Jacob Jacobi (1804–1851) and William Rowan Hamilton (1805–1865). Hamilton’s approach is es-pecially revealing. He introduced what seems only a small modification to theEuler-Lagrange equations, taking as his characteristic function the sum of the visviva and the potential, H = T + U, where, as for Lagrange, T is a function ofboth position and momentum but U, the potential, depends only on position. Thisfunction could be transformed into the Lagrangian by a simple rule, using the gen-eralized conditions that the force—in the original sense of a change in momentumwith time. Hamilton also wrote the velocity in a new way, as the change of thischaracteristic function with momentum. The surprising consequence of this purelyformal change in definitions is a new framework for posing mechanical questionsthat is completely consistent with the least action principle. The equations of


motion become symmetric: a time change of position depends on the momentumchange of the function H and the rate of change of momentum with time dependson the spatial changes in H. This function, now called the Hamiltonian, is con-stant in time but varies in space; the motion is, therefore, constrained to maintainconstant total H. This is a result of his choice of transformations between the coor-dinates but there was something more here that Lagrange had not realized. Thesemethods were remarkably fruitful, so much so that in the twentieth century theywere the basis for the development of quantum mechanics and field theory, as wewill see. They also served for celestial mechanics in an important way. They permita transformation between measured properties, such as angular motion, to moregeneral properties of orbits, such as angular momentum. At the time Hamilton wasconcerned with optics, specifically how to describe wave propagation mechani-cally. For a simple wave, the frequency and phase of oscillation provide a completedescription of the force. For light the propagation speed is already known and herealized that a plane wave is defined by a surface of constant phase. He identifiedthis phase with the action, he called this the eikonal, and by assuming that thetrajectory followed by the wavefront is that which minimizes its change in phase,recovered the least action principle.

The utility of the Lagrangian approach, and the later generalization by Jacobiand Hamilton, is its transparency when changing from one coordinate system toanother, especially when there are constraints. This had already been introducedin celestial mechanics by Poisson. He realized that instead of working with thedynamical quantities, position and velocity components, it is more natural to rep-resent orbits in terms of measurable geometric quantities such as the eccentricity,orbital period, and eccentric anomaly. He showed how to transform the equationsof motion using these parameters and his mathematical device was essentiallythe same as Hamilton and Jacobi found for passing between different referenceframes and conserved quantities. There are a few examples that will show how thisnew approach could provide answers to mechanical problems that otherwise werequite cumbersome. One of the most beautiful comes from examining a device thatwas used to illustrate ideas of equilibrium of forces, introduced as a schematicmachine by John Atwood in 1784. It is the simplest machine, the descendent ofJordanus’ idea of positional weight except in this case the masses are connectedby a cord draped over a pulley (whose mass we will neglect) hanging aligned andvertically. If the two are equal, they can be placed in any relation with respect toeach other and if one is set into motion, the other moves in the opposite directionwith precisely the same speed. If, on the other hand, the masses are unequal, theycannot be placed in any relation that will allow them to remain stationary.

INTERMEZZO: THE STORY SO FAR

Let’s now sum up the developments of the force concept at it stood by the middleof the nineteenth century.

For Newton himself, the pivotal moment came with the realization that, ac-cording to his conception of force, any deviation from inertial motion must be inthe same direction as the force provoking it and of an equal magnitude. Thus, forany mass, any change in the path is the result of an external source producing its


acceleration. If a mass falls, it proceeds toward the center of the force. This means,here, toward the center of the Earth. because, then, the Moon orbits the Earth andthere is nothing else changing its motion, it must be able to fall precisely as a rockat the surface. Being farther from the center, the influence of the Earth—the forceper unit mass and thus the acceleration—will be smaller but the same terrestrialrock would behave just like the Moon at the same distance from the center. Thatthe motion in angle is different is easily dealt with. A rock thrown horizontallywould, asymptotically, simply never hit the surface as its “inertia” increases, itwould continue to fall without ever hitting anything. If the Moon, in this samesense, were to stop in its motion, it would fall and take the same amount of timeto reach the center as it does to orbit because the freefall time is the same as theorbital period. Again, there is no angular forcing, no torque.

Now for the next step, we say that the only agent producing this is the masscontained within the orbiting path. Any outside has a perturbative, deviating effectunless it is identically distributed as the central body. In the symmetric case, thereis no additional influence, apart from some possible viscosity (if this is a fluid,for instance), only its mass within the circle enclosing the orbit would directlyaffect the orbiting body and this identically as if all of the interior mass wereconcentrated at a point in the center. The power of this realization was immediate:all bodies orbit the Sun following the same law because it is the dominant force.For the Earth–Moon system, the Earth has the dominant influence on the Moon,as on the Rock, but the two orbit the Sun because the differential force of the Sunacross the lunar orbit isn’t sufficient to make the Moon also orbit only the Sun.This produces, however, a phase dependent force, one that changes throughout themonth (lunar orbit) because it changes phase through the year. The same for thetides, the differential force across the Earth is large enough to produce a relativedisplacement that depends on phase for the two sides of the Earth.

The dynamical theory of tides then takes the inertial reaction of the masses intoaccount, allowing for internal forces to change the motions, but the static theorywas completely developed subject only to the distortion produced on the body inequilibrium. Allowing for a different reaction time for the fluid and solid parts ofthe body produces phase shift between the force maximum at any point and thereaction. This was the step taken by Laplace.

Newtonian gravitational theory was, however, only concerned with the forcesand therefore the change in viewpoint with the potential function was to adda means for deriving the forces from the mass distribution. Independent of thesymmetry of the body, the forces were locally produced by changes in the potentialalong the direction normal to any surface. The geometric form of the body thusproduced a concentration of the force, or a reduction, depending on the gradientof the field. Again a geometric device replaced a physical explanation, it doesn’tmatter what the field is, any field will behave identically if the force is givenby a field. The symmetry produces certain constant quantities, for instance ina spherically symmetric or axially symmetric system the angular motion remainsconstant because there is no gradient in the potential along the direction of motion.But if there is any change in the form of the body, a broken symmetry, that altersthe angular as well as radial motion and the orbit will no longer close. The motionof the body, whatever the source of the field, will alter.


The last step was to take this static picture, where the masses or charges werenot changing their positions in time, and allow for the source itself to vary. Inparticular, if the geometry of the source changes in time, this produces a changein the reaction in time. For a gravitational field this is a gravitational wave. For anelectric field, a charge distribution, this is an electromagnetic wave. Mass currentsare different than charge currents but not so different at this basic level. A wave isa time dependent acceleration at a point in space. And the inertial response of abody depends on its coupling to the time variation, if there is an angular componentto the wave this produces an angular response, if it is strictly radial this producesa simple harmonic oscillator in radius and the body doesn’t displace relative to thecentral source. Thus, for Newtonian gravity, there is only instantaneous change,the effect of introducing a spacetime is that all changes require a finite periodand thus resemble electromagnetism. There are waves, polarized and accelerating,for gravitational fields exactly as for electromagnetic and because currents andtime dependent displacements of charge are of similar character, electromagneticwaves require both changes in the magnetic and electric fields.

NOTES

1. For the rest of this book, we will describe this as a “scalar” quantity, something thathas only magnitude, for instance a length or speed. In contrast, a “vector” quantity has bothdirection and magnitude, such as a force or a velocity. The terminology was introduced byHamilton but the ideas are as old as the composition of motions and the intension of forms.

2. To say this in more precise language, we take the field strength to be the gradient ofa function, � that depends only on space so that E = grad �. Then the divergence of thisquantity, summed over a completely enclosing volume, is

∫div EdV =

∫E · nd A.

Here V is the volume for which A is the bounding area and n is the direction of the surfacenormal. The divergence is scalar in the sense that it is a quantity, a rate of change in alldirections of something, while the gradient is a vector since it is the rate of change alongeach direction. By dimensional analysis the field strength varies as the inverse of the areatimes the charge within that distance so the volume integral of the density is the totalenclosed charge. This results in the Laplace equation for a point charge and the Poissonequation for a distribution.

3. Let me advertise the discussion of relativity theory in Chapter 8. You’ll see why thisway of describing a field was so important for Einstein during the first stages of his gropingfor a way to express a generalized theory of gravity using curvature, hence a geometricframework, is natural for the description of the field. His use of the word potential, or field,for the metric—the way to measure the structure of the space—is not a casual choice:if you think of the potential as a surface, and the motion is therefore expressed in theextended framework of a four dimensional world instead of only three, if the time andspace are not really independent of each other, then the acceleration can be understoodin terms of gradients of those surfaces and in terms of their curvature.

4. This work is famous for its preface, which declares “One will not find figures inthis work. The methods that I expound require neither constructions, nor geometrical ormechanical arguments, but only algebraic operations, subject to a regular and uniformcourse.” With this treatise, Lagrange was continuing the mathematization program begun


by Galileo but with a radical modification. While previous arguments had used geometricformalism and presentation, the Mecanique Analytique explicitly adopted purely formallanguage. This approach was to take root on the Continent rather quickly, it took severaldecades of propagandizing in the first decades of the nineteenth century by CharlesBabbage, John Herschel, and George Chrystal in England to affect the same reform.


6

THERMODYNAMICS ANDSTATISTICAL MECHANICS

Birds of a feather flock together—English proverb (c. 1545)

Science and society converged at the end of the eighteenth century with the be-ginning of the “‘Industrial Revolution,” the development of factory-based massproduction of identical goods at high rate and of almost unlimited variety. Me-chanical invention provided the impetus for this momentous shift from small scaleto industrial mass production and with it came a new concern: efficiency. Centralto this change was the steam engine, in its manifold forms.

The start was innocuous enough. Coal recovery was frequently hampered byflooding, a problem that could be alleviated simply when the mines were shallowby ordinary air pumps acting against atmospheric pressure. But there is a limit tosuch work, a column of water cannot be raised in single steps by more than aboutten meters. Galileo, Torricelli, and Pascal had known this two centuries earlier.Georg Agricola, at the end of the sixteenth century, had discussed its practicalconsequences. For deep pits, many pumping stages were required to elevate thewater. To make the process more efficient, some force had to actively drive thefluid to greater elevations. This required large pumps and the most efficient driverwas a battery of steam engines.

The second was the problem of factory layout and operation. One skilled artisanworking through a complete operation was too slow for the industrialists. Instead,the separation of individual tasks required specific machinery for each and theability to pass work partially completed to another hand for finishing. Water powerwas enough for the older collective producers. But for the expansion of scaleneeded to satisfy ever-increasing consumer demands, a more machine-centeredmode of production was needed. Water and wind power, both of which sufficedfor smaller scale manufacture, face severe limitations when scaled up to the levelrequired for the new factories. There is no power to drive a wheel if there is no


flow; a dry season could be a financial disaster. A wet season, on the other hand,could prove equally devastating since often there were no reliable mechanismsfor controlling the rate of flow. This hydro-power also required mills to be placedat specific locations. Further, the power could not be easily distributed within alarge factory complex.

The industrialization of Europe was advancing apace and increasingly requiredtechnological improvements to improve the efficiency and reliability of its machin-ery. In this context, the growth of thermal physics was a natural, almost inevitableoutgrowth of a social transformation. The new questions raised by the needs of theindustrialists provided fruitful and challenging problems for the mathematiciansand scientists of the era. In particular, in the context of the Newtonian programand philosophy of the regulation of natural processes by natural laws, there was anopen territory for exploring new theoretical and applied ideas. Human labor wasdifficult to quantify and even harder to control. The machine provided importantbenefits to the new “captains of industry” and the capitalists who financed them.One was the ability to precisely measure the efficiency of the process, the amountof fuel required (in whatever metrical units one would want, for example cords ofwood or tons of coal) and some measure of the productivity or output work, forinstance the amount of cotton separated and combed or the area of woven cloth.As the devices, mainly steam engines, were put into progressively wider service,the measures of the work achieved varied as did its physical manifestations. Over-all, however, it was becoming clear that Nature itself operates with some sort ofbalance sheet, totaling the costs and production to lead to profit.1

There are many examples, I’ll choose just a few. The expansion of steel pro-duction was spurred by large projects such as the construction of metal ships,bridges, and buildings. While much of this could be carried out using purelypractical techniques, such as those used by the medieval master builders in theconstruction of the Gothic cathedrals, national-scale projects such as the creationand expansion of railroads during the first half of the nineteenth century through-out Europe and North America and the hydrological studies required during theera of canal construction and the increasing needs for water supply to expandingcities, all produced auxiliary theoretical problems of generalization and extensionthat dominated much of physics during the nineteenth century. It’s important tokeep this in mind. The expansion of physics, chemistry, and mathematics duringthe century was frequently spurred by complex, practical problems that requiredgeneral laws rather than “rule of thumb” approaches.

Above all, the invention of the steam engine was a fundamental historical turningpoint in technology and physics. While during the previous centuries ballisticshad provided a (unfortunately frequent) context for posing general questions indynamics, the newly founded centers of industry, whether Manchester in Englandor Clermont-Ferand in France or the Saar in Germany, also required a growingpresence of technically trained engineers and designers. This in turn required anexpanded university presence for science, especially physics, and it was duringthis period that many new universities were formed, and the curricula changed toemphasize the teaching of physics at all levels of schooling.

The steam engine provided the framework in which to explore what was quicklyrecognized as an obvious connection between heat and force. One way to think of

Thermodynamics and Statistical Mechanics 119

how the picture changed is to recall a level or a block and tackle. These simplemachines converting in effect, one form of force into another and as such had beenclassified by Hero of Alexandria and Vitruvius soon after Archimedes. But a steamengine is very different. It transforms heat into work, taking energy—not simplyforce—from the state changes of a hot vapor and converting that to mechanicalaction with some efficiency that can be quantified. Although the essential forcelaws for vapors and liquids had been explored in the seventeenth century—forinstance, Boyle’s law for the change of pressure in a gas on compression—thethermal contribution had been neglected. Even the atomists, for instance DanielBernoulli, had pictured a state of agitation of microscopic particles that didn’tinvolve a change in their momentum distribution; the particles merely impact thewalls more frequently when the volume is decreased but that’s purely a densityeffect. Instead, to consider changes when a quantity of heat is added, was inthe program of the nineteenth century and was spurred by the development byNewcomen and Watt of engines capable of driving pumps, lifting heavy masses,and running whole factories.

It was a development in part directed by an economic need. A steam engineconsumes a quantity of fuel. This costs money. It produces a measurable output,work, that represents a certain income. Far more than human or animal labor,where the fuel source is far less immediately understood, anyone can understandhow much petroleum or wood must be burned to produce a specific output of themachine. Thus was formed the research program of thermodynamics. Heat is aninternal property of an object, something that can be quantified with a thermometer(any device will do). But work is the measure of its interaction with an environment.A piston expands to push a level to drive a wheel. The momentum of the wheelis the result, then, of a force originating at the piston. The fuel, steam, changesstate—it cools—thus releasing a certain amount of received heat, supplied bya fire, to a motion. Yet state changes were also involved. A column of liquid,mercury or water, expands when heated. That is, simply imagining a verticalcolumn standing on the surface of a table, the change in the height is a work doneby changing something internal to the fluid. The implication that something hasbeen transferred to the fluid from the surroundings, perhaps a fire, that withoutany distant action changes the state of the material. But how do you define theefficiency of any device for converting energy—here in the form of heat—intowork, and how do you describe physically what happens.

The dream of the industrialists was to get something for nothing, to get more workout of their machines than the fuel supplied. Motion once started might continueforever, that’s what inertia means. But real devices have dissipation so, given theselosses, is perpetual motion—or better, perpetual action—possible? As early asthe late sixteenth century Simon Stevins had considered the perpetual motion ofa continuous chain hanging under its own weight on an inclined plane. He askedwhether this “machine” could continue to move or, for that matter, would startmoving, abstracting the essential dynamics from this rather cumbersome driving.His idea was to contrast motion on an inclined plane with freefall, hence thechoice of a wedge rather than something more realistic. The answer was no, andthe same reasoning applied in Galileo’s discussion of inertia. These were relativelysimple to address since the equivalence of actions of weight independent of the


constraints of the geometry. But when it came to mechanisms driven by heat theanswer wasn’t so clear.

CHEMICAL FORCES

To understand how to transform matter into work required knowing what forcesgovern its states and reactions. For Newton, chemical combinations and trans-formations served as microscopic analogies to the forces and their transmissionon large scale. But his speculations, such as those appended to the Opticks be-tween editions of the Principia, were far from quantitative. In the last third of theeighteenth century and the first decades of the nineteenth, chemistry became asystematic science in the hands of Antoine Lavoisier and John Dalton. Chemicalsystematics were discovered, along with the existence of substances that despitedistillation could not be reduced, the elements. Lavoisier showed that oxygen isthe active agent releasing heat in chemical reactions in air and Dalton discoveredthe combinational rules for weights of the elements. With the demise of a specialfluid to explain combustion and the discovery of integer relations among elemen-tary volumes and weights of the elements, chemists turned again to the atomicconception of material structure.

Recall that although the atomic picture was a ancient inheritance, it came alongwith an antique mechanism to explain the structure of matter. All forces werecontact and specific to the nature of the particles: hooks and eyes, sticky surfaces,and analogies familiar from the artisan’s workshop. As pure analogies these workedperfectly to “preserve the appearances” but they were useless for predicting newphenomena. Even more extreme was the hard elastic sphere used by Boyle andBernoulli to explain the gas laws. Again, the hypothesis explains the observationsbut the model leads does not suggest new phenomena. Transformations of thesort so familiar within alchemy and later in experimental chemistry requireda new approach and this was provided by Boscovich in his Theory of NaturalPhilosophy (1758). In the spirit of the mechanical philosophy, he introduced asort of field theory into the atomic world, picturing the interactions among atomsas both attractive and repulsive according to the specific elements involved. Thiswas not just a hypothesis. By the middle of the eighteenth century it was clearfrom generations of exploration that there are a number of substances that, despitebeing subjected to continued treatment with heat and distillation, cannot be furtherpurified. These were called elements and the study of their properties founded notonly modern chemistry but also renewed the atomic view and brought the study ofmatter into the proper domain of physics.2

In one of the most fruitful scientific collaborations of the eighteenth century,Laplace and Lavoisier studied the properties and chemical reactions of solutionsand introduced some of the basic concepts that later dominated chemical thermo-dynamics. They imagined immersing a beaker in which two solutions had beenmixed within a huge block of ice, or better a thermos-type bottle in which insteadof a vacuum the space between the two cylinders is filled with packed ice. Whenthe reaction occurs heat is released. This is transmitted to the ice, which melts,and the water is drained. By weighing the quantity of water, knowing the specificheat, the amount of energy released can be determined. This is a system that


changes energetically but not mechanically, in effect this is a system that does nonet work but changes its internal state, its heat content. Importantly, at this statechange isn’t reversible. Another example is to think of a rock. You lift it and dowork on it against a potential field, the background gravitation. When released itwill have gained no net energy, no net work can be done on the system. But whenit hits the ground some of the potential energy is un-recovered, it goes into heatingthe body, perhaps breaking it. You may be able to use the rock later, if it’s hotenough (try to imagine that) to heat something else but it will not spontaneouslycool and jump into the air. In both cases, we’re really confronting the earliestconcepts of potential versus kinetic energy. But the two, and many others, arealso illustrating the difficulties of maintaining such motion under less than idealconditions. If there is any friction, between rubbing solids and surfaces or internalfluid viscosity, the motion will clearly cease.

MACHINES AND THERMAL CYCLES

The idea of using heat and steam power as a driver is as old as the Greeks. Heroof Alexandria described a wonderful device for opening temple doors when asacrificial fire was lit on an alter. But the development of a machine driven byburning coal or wood to heat a boiler and drive the expansion and contractionof a piston was the product of the eighteenth century and changed the world. Itbegan in those same mine shafts I’d mentioned a moment ago. As the need forcoal grew, miners descended ever deeper and the problem exacerbated. The firstpractical steam powered engine, invented by Thomas Newcomen in 1712, was putinto service as a pump. The difficulty with the original design was a tendency ofthe machine to go berserk, to overdrive and rupture. The valved condenser anddynamical control were invented by James Watt (1736–1819) in 1769. For thecondenser, the expansion and evacuation were moved to a separate container.For the control, he attached a spinning ball, suspended on flexible supports, toa shaft that was, in turn, connected to the driving shaft of the main wheel of theengine. In this very neat mechanical invention, if the wheel turned too quickly theballs rose and extended, increasing the moment of inertia and slowing the devicemuch as a skater does in extending his arms. If the rate fell, the balls droppedand contracted below their balance point and the centrifugal force reduced. Thisfeedback mechanism, called a “gouvernor” for obvious reasons, made steam powerpractical. Thus the steam engine became the paradigmatic device to illustrate thelater discussions of dynamical thermal processes. While the concept of heat wasstill ill-formed, the load of wood (and later coal) needed to fuel the engine made itclear there was no perpetual work. But the discussions of the mechanisms were stillconfused. While force was obviously produced because the pump was clearly doingwork in lifting a weight (in this case of the liquid), and this came at the expenseof the supplied heat, it was far less clear that there was actually a conservationprinciple here analogous to that of momentum.

HEAT AND WORK

A force can be produced by, or can produce, changes in the “state of a body” thatwe call heat. How this internal property is changed into an external acceleration


and/or work, is something that perplexed the physicists of the nineteenth centuryand still produces confusion. You see, the motion of a particle being acted on byan external force is a comparatively easy thing to follow relative to the change inthe condition of the body or system being thus affected. Work, the production of amechanical effect, is something that can be quantified but the relative balance ofenergy among different components, or types, is much more difficult to ascertain.For instance, how much heat is added to a body can be measured only if you cancontrol the state of the system at a particular time.

Let’s stay with the steam engine for a moment (although you could imaginethis in any generator, the treatment of a nuclear reactor is virtually the sameexcept for the fuel). The operation is something like this, in general. Place acylinder filled with water in contact with the heat source, a coal fire for instance.Absorbing or releasing heat is a different from doing work. When in contact witha heat source, the gas temperature rises in the cylinder but there is no work. Anyaddition of heat produces both a change in the internal energy. Now take the bodyaway from the heat and allow it to expand. This will produce the work since thecylinder, if properly constructed, drive a lever and through that anything else inthe outside world. The machine does work. But there is a price. The internal energydecreases, the cylinder cools. Now, once we reach a specific temperature, we stopthe mechanical process and allow the rest of the heat to be released, perhaps eveninvolving flushing condensed water out of the cylinder. Then we compress the gasfurther in isolation, and prepare it to be again placed in contact with the heat source.

A theoretical framework that defined the efficiency of a thermal engine wasfirst proposed by Sadi Carnot (1796–1832) in his 1824 memoir “On the MotivePower of Fire.” Carnot’s approach was to consider a ideal machine, an abstractdevice without considering its construction or composition, operating between twofixed temperatures. These determined the quantity of heat transferred to it from areservoir. In this schematicized picture the bath in which the device is immersed,or its sources and sinks, are irrelevant to the operations. Its sole function is toconvert a difference in temperature, produced by the absorption of heat, DQ, intomechanical work, W . As a measure of the efficiency, he proposed the expressionW/DQ, the work done per unit heat absorbed. He imagined a reversible process,or a sequence of them, that not only operates between fixed states but occurs soslowly that the device can remain in equilibrium at each point in its changes.By this Carnot meant again to draw the comparison with a steam engine but onethat moves so slowly that at any moment the entirety of the device is at the sametemperature and there are no net accelerations in the system. In this case, duringthe interval when things are changing, there is no exchange with the sources orsinks. This occurs only at the ends of the process. For a cylinder of a steam engine,this is a cyclic change. The system absorbs heat, does work, and them preparesitself by a mechanism to again receive some quantity of heat. This is now calledthe Carnot cycle. The cycle can be subdivided when operating between any setof temperatures such that the extrema represent the global change in the systemwhile the subcycles can be considered as taking the system between a set ofintermediate states each of which is only virtual.

Carnot used indicator diagrams to illustrate the process, a graphical deviceintroduced by Watt to describe the cyclic operation of the steam engine in which


the state of the system is described by two easily measurable quantities (such astemperature, pressure, and volume). Various combinations were employed, de-pending on what properties were essential for describing the machine’s operation.One could choose these intensive variables (those not subdivided within the deviceitself), the temperature and pressure were the principal physical quantities, andthe volume of the device since the work of a steam engine is done by successiveexpansions and contractions. There is a stage when condensation occurs becauseof the expansion, the part of the cycle that ultimately limited its work, becauseof the collapse of the steam to liquid when some critical temperature is reachedon expansion. But that was expelled from the system, a waste product, which iswhy the machine cannot be completely efficient. The beauty of this abstraction isthat you can ignore the structure of the device. In fact, any thermo-mechanicalsystem, no matter how complicated, can be treated as one composed of parts incontact. This was later generalized to include the idea of a heat bath, a completeenvironment in which a body is immersed, and made more precise the idea ofequilibrium.

To then extend this approach we note that it must be true for any system if thepath of the transformation doesn’t matter so we are free to choose the description ofthe material. Benoit-Pierre-Emile Clapyron (1799–1864) and Rudolph Clausius(1822–1888) chose to quantify the effects using an ideal gas, one for which thepressure depends only on the temperature and volume. Then because the internalenergy of the whole cylinder depends on the temperature alone, the transformationof heat for a reversible process is a smooth one that has the change produced bythe addition of a quantity of heat to a thermodynamic system produces work andchanges the internal energy. This is the first law of thermodynamics, the relationthat defines the mechanical content of thermal physics. Clausius, in particular,distinguished two energies of any mechanical system. One is the center of massaction, the bulk motion generated by the potential or force. Whether this is apressure acting on the device or a force on its center of mass doesn’t matter. Butto expand this definition meant accounting for both external and internal changesand, within a body, the energy is quickly (it was assumed) redistributed in a widespectrum of motions, none of which can be individually resolved. None of thesechanges produce any net effect on the body as a whole, it continues to move asit had although its shape may change because of internal rearrangements. Thusthe center of mass acceleration represents the reaction to a potential, while thechange in the internal energy—what Clausius described with his title—is “a kindof motion we call heat.” While it is possible to use this waste product of theaverage mechanical process to do other things later, it isn’t available to performthe original function of the device.

HEAT AS A FLUID

Steam engines and other thermally driven machines are conversion devices takingsomething from the fire and converting it into work. There was an analogy herewith Franklin’s conception of a single fluid for electricity (although this had beensuperceded by the end of the eighteenth century, as we’ll discuss in the nextchapter), a thermal fluid that is transferred from something having an excess of the


fluid to something else that’s deficient. To make the analogy clearer, we imaginea quantity of something we’ll call heat, caloric, that can be transferred preciselylike a fluid: it is assumed to be infinitely divisible (not particular) and is measuredby something we will call temperature. To say something is hot, instrumentally,is to take a reading with a device—a thermometer—and notice that the readingis larger when we think the body is hot, as Carnot says, which is how there ismuch heat in a substance. While we really don’t yet know what this means, we canimagine changing the amount by DQ. Then there are two effects and this is thefirst law of thermodynamics, that the quantity of heat added to a system in contactwith a source produces a change in the internal energy plus some work done by thesystem on its surroundings (or anything else to which it’s mechanically coupled).This is a law, on the same footing as Newton’s first law defining inertia. As longas a body is in contact with a source of heat, work can be driven but the body willalso get hotter.

Now in keeping with out definition of work as the action of a force during adisplacement by that component lying along the direction of motion, we can alsoimagine an extended body changing. Here we don’t ask detailed questions aboutthe internal structure, simply that there is a way of describing how the body as awhole will react to the addition of this heat. The body has a volume (or a length,this argument applies equally to the change in the length of a rod, for instance, ifheat is added) and, therefore, a surface, and the pressure is defined by the forcedistributed over an area. If we sum over all the areas and take the increase in thevolume to depend only on a lengthening of the column (imaging a cylinder, onlyone end changes while the walls remain rigid). Then the work, which is the forcetimes the displacement, becomes W = P �V for some infinitesimal change inthe volume, �V if the pressure is constant during a period of very small changein the volume. We’ll return to this point in a moment. Remember that Carnotimagined that the changes are taken slowly and in infinitesimal steps so there issufficient time in each step to reestablish mechanical equilibrium. This requiredthat the input of heat produces a change in the internal energy E and the workW (DQ = �E + W ); E is that part of the heat that doesn’t produce work or achange of state of the body. The way we change the system matters, and this is wherethermodynamic reasoning is so different from ordinary dynamics. If we hold the vo-lume constant, or we hold the pressure constant, we get different results for thechange in the internal energy. If we consider the thermal analog of mass to bethe heat content divided by the temperature, �S = DQ/T , then we can imaginethe change in the heat being a simple scaling number. If the heat capacity of abody is constant then DQ/T is independent of temperature and therefore shouldnot vary if the changes occur slowly enough that no net work is done, an adiabaticchange. Now let’s go back to Carnot’s hypothetical machine that simply operates,somehow, between two temperatures, and ask what amount of work we can getout of the device efficiently. Let the quantity of heat initially absorbed be Q1 at atemperature T1. Then let the system expand without loss of heat. Then imagine thesystem to be cooled, or to perform work—it doesn’t matter for our purposes—butthe amount of heat lost should be the same as the amount gained except that itoccurs at a lower temperature T0. Then let the system contract, somehow, untilit is brought again into contact with the heat source. The total work done will be


proportional to (T1 − T0)�S , the amount of heat absorbed will be DQ = T1�Sso the efficiency will be η = (T1 − T0)/T1 < 1. In finding this result, that it isimpossible to create a perfect machine, Carnot demonstrated conclusively thatperpetual motion cannot exist even for a purely thermally driven system.

MECHANICAL EQUIVALENT OF HEATAND HEAT CAPACITY

The specific heat is defined as the coefficient for the change in the internalenergy—that is, the temperature—when a quantity of heat is added but what “theamount of heat” means was ill-defined without some relation to work and/or forces.This problem was solved by determining the mechanical equivalent of heat. Theconcept originated with Count Rumford (Benjamin Thompson, 1753–1814), asoften happened in the early history of mechanics, through a practical observation.You know that rubbing your hands together produces, through friction, a feelingof warmth. Repeated bendings of spoons produce substantial heat and even even-tually leads to it breaking. Stressing a rubber band between your lips will producea sensible temperature difference. Rumford noted that an enormous amount ofheat was generated by friction in the boring of cannons, and communicated hisobservations to the Royal Society in a series of influential memoirs. These commonexperiences make you suspect there is some connection between work and heatbut aren’t quantitative. That was accomplished experimentally in an elegant wayby James Prescott Joule (1818–1889). He inserted a paddle-wheel in a water-filledsealed cylinder and connected the mechanism to an external hanging weight. Heused distilled water, which was the standard substance in the nineteenth centurybecause of its stability at high temperature and the ease with which it is purified.The centigrade temperature scale was calibrate by Celsius using its two well de-fined phase transitions: when pure water freezes and when it boils. Joule let theweight descend and measured the temperature rise of the water. The work donein dropping the weight depends only on the distance through which it is moved,not the path or the acceleration (which is always gravity). Once you have thisquantity, the specific heat can be defined as the rate of change of heat content withtemperature, C = DQ/�T . The precise value of this depends on the substance.But how to relate it to the properties of matter was the beginning of a separatestudy, the statistical description of motion on the microscale.

EXPLAINING THERMODYNAMICS:STATISTICAL MECHANICS

Thermodynamics was about dealing with heat, energy, and work. For all the ther-modynamicists knew, these could be fluids that are exchanged between elementsof any complex system. The chemists were, instead, more concerned with com-binational regularities and to them the world was increasingly obviously atomic.Mineralogy also revealed the connection between forces and structure at the mi-croscale. While atoms in motion are difficult to imagine, and even harder to dealwith theoretically because of the statistical element of their interactions, crystalstructure is a clean example of how the forces between atoms can be understood.Different elements, carbon for example, structure themselves in a limited set of


structures, the lattices that echo the Platonist vision of the “ideal, geometric” un-derpinnings of physical laws. Crystals display geometric regularities that fell intoa small number of point group symmetries that also seemed to require some sort ofmicroscale discreteness. Finally there were phase changes in matter, transitionsbetween solids, liquids, and gases, that required or released heat accompanied byoften dramatic changes in volume without changes in weight that were consistentwith an atomic model. These growing conflicting views about the nature of heatreflected a more fundamental shift in dealing with matter itself. The interactionof atoms among themselves was, in Gibbs’ view, the reason for phase changes. Achemical potential measures this and even in the absence of motion and at bothconstant pressure and temperature, reactions change the entropy of the systemby altering the number of particles. An increasing collection of observations andexperiments were, by the last third of the nineteenth century, pointing to bothan atomic view of microscale interactions and a discretize picture of the inter-nal structure of those particles. These realizations arrived from many differentdirections, some from the laboratory, some from natural phenomena, some fromtheory.

The Virial Theorem and Phase Space

Thus, during the second half of the nineteenth century, heat ceased to be a fluid,or a field, or something substantial. It became, instead, a dynamical property ofmatter. It was something within things that connected the structure of materialwith energy. Since matter and motion, for the previous 150 years, had becomeprogressively more unified, the stage was set to add thermodynamics to this. Butit was clear that something also had to change regarding the usual mechanicalideas for interactions of masses. Thermal properties are not universal in the waygravitational interactions are. Different compositions produce different behaviors.Even for the same material, depending on its state the thermal properties aredifferent. The gas laws and chemical systematics discovered Lavoisier, Dalton, andtheir contemporaries between about 1780 and 1820 seemed to demand an atomicview of matter and with it came the question of whether the dynamics of atoms aredifferent from those of macroscopic bodies. How is this microworld structured?This became a central question of physical theory in the middle of the nineteenthcentury. The laws governing the motion of atoms were constructed analogously tothose governing gravitationally interacting masses. Although gravitation plays norole, and this was quickly realized, the idea that interactions can take place at adistance through the action of a field dominated the modeling effort.

The modern concept of statistical mechanics was founded in the second halfof the nineteenth century, mainly by James Clerk Maxwell (1831–1879), LudwigBoltzmann (1844–1906), and J. Willard Gibbs (1839–1903). The virial theoremof Clausius was an important step in this program. Although it was implicit in thedynamics of macroscopic, gravitationally dominated systems of masses, Clausiusrealized that it could be generalized and applied to any system in equilibrium. Herealized that since the combined potential and vis viva—kinetic—energies takentogether are the total energy of a body, he extended this concept to a system ofbodies mutually interacting through any potential field. The resulting statement


is that the sum of all kinetic energies K of the particles was half of the totalenergy, or that the binding energy of the system (in equilibrium) is half of the totalenergy. This is for a linear system, one where the potentials depend only on therelative positions of the constituent particles, written as 2K + = 0 if the systemchanges so slowly that changes in its moment of inertia can be neglected. This isa global constraint on the system, the trajectories of the individual constituentsdon’t matter. Joseph Liouville (1809–1882) had derived a related constraint thatfor an adiabatic change, the distribution of particles would shift around in spaceand momentum and exchange energy but maintain constant energy.

This is why the dynamical picture developed by Lagrange and Hamilton wasso important. Every particle at any instant in time has a defined position andmomentum. Each of these has three spatial components, since they’re vectors.Taken together for N masses they form a 6N dimensional space, called a phasespace. You’ll recall this from the discussion of celestial mechanics, orbital motionsand perturbations are described in terms of orbital elements that are functions ofthe energy and angular momentum and the equations of motion are the rate ofchange in time of the positions and momenta with changes in momentum andposition, respectively, of the Hamiltonian. For a gas, we eschew the possibility oftracing the motion of any single particle and instead assert that we only measuremean quantities of a statistically distributed set of positions and momenta. Weimagine a collection of masses each occupying at any instant a point in this spacebut, because of collisions and distant encounters, the coordinates randomize andsay they behave probabilistically. What we see is an average over some statisticaldistribution, we’ll denote as f(x, p, t), which is the probability of finding anysingle mass in the ensemble at a specific point in phase space. For a uniformdistribution in which no spatial coordinate are correlated, we can write f(x, p) =f(x, p1)f(x, p2)f(x, p3) = n(x)f(p), where n is the number density.3 Then if themomentum is identically distributed in all directions and the total kinetic energyis the sum of the partial energies (that is, = (p2

1 + p22 + p2

3)/2m for identicalparticles of mass m) there is only one function that satisfies the requirements. Thisfunction was found in 1859 by Maxwell in his Adams prize essay on the stabilityof Saturn’s rings and made more precise in his papers of 1860 on the dynamicaltheory of gases: it is the Gaussian or “normal” distribution that was alreadyknown from statistics, f (p) ∼ exp(−p2/(2mkT ) in an interval of momentumdp1dp2dp3 = 4πp2dp (since each interval of momentum is also identical) wherek is a constant, now named after Boltzmann. Thus, if the energy is constant in thesense that the collisions do not produce losses from the system, we can changethe average from the relative motions to the individual energies and write thedistribution in terms of the kinetic energy f (E). Each particle has a kinetic energyof kT /2 for each degree of freedom.4 We can say nothing about the individualtrajectories but we can make statements about the ensemble and these are thethermodynamic quantities. The biggest change in viewpoint is the reversibilityof the action and reaction and the difficulty of assigning an order to the effects.Instead of solving the equations of motion for a single body, we imagine a collectionthat spans the full range of possible positions and momenta but nothing more. Theequations of motion for macroscopic bodies can be explained by this picture. Theatomic world of this classical physics requires more a shift in focus. Our interest


is now on the interactions and not the trajectories, on how the forces that governthe microworld average to produce the observable behavior.

Brownian Motion

A glimpse of the microscale was provided by a simple observation by RobertBrown (1773–1858) when he described the motion of pollen grains in a liquidviewed with a microscope. The grains moved unceasingly but irregularly as ifalive. This was not entirely unexpected, impulses had been used since Newton todescribe how force is transmitted to the mover, but now it was different. Insteadof the deterministic motion generated by a central field, this motion is random.This was Albert Einstein’s (1879–1955) first fundamental contribution to physicalscience. In a series of papers in 1905, he solved the problem of how an atomicscale process can produce visible displacements. He assumed the cause of themotion is a continuous, enormous number of infinitesimally small impacts by theatoms of the medium in which the larger particles were immersed. Each impactcan be in any direction and over an unbounded range of strengths, but the mostprobable are vanishingly small. To account for this, he assumed a Maxwelliandistribution for the velocities and showed that the motion describes a diffusionequation by treating the motions as fluctuations of a continuous property, thechange in the momentum of the particle. Each impact produces a deviation. Ina random distribution of changes in velocity, the most probable value for �v inany direction is zero. But there is a dispersion of the fluctuations and this nevervanishes. For an ideal gas, as we have seen, it depends on the temperature. Thisprediction led to series of delicate and beautiful measurements by Jean Perrin1870–1942) in 1909 that traced the trajectories of particles and quantitativelyverified Einstein’s law. Paul Langevin (1872–1946) generalized this by writingthe equation of motion for a random acceleration. Again, he assumed the meanstep size is zero but that the random accelerations don’t vanish because theirdispersions remain finite. For Newton this would not have seemed at all unusualexcept for the assumption that the impulses are random, and perhaps not eventhen. But to a community used to thinking of forces as continuously acting, thistreatment was strange indeed.

We only need to think of a random walk as a set of discontinuous steps. At eachmoment, t , we have a distribution of kicks, f (�p). To make this even simpler,think of this distribution as the probability of sustaining an impulse �p that israndom in both magnitude and direction and there are no correlations. Anotherway of saying this is there is no memory in the system and each impulse is strictlyindependent of every other. Then we’re not really computing the motion of any oneparticle but the collection of trajectories and accelerations of the whole ensemble.Each has a separate history. In probability theory this is now called a Markovprocess, where the probability of being at a position x + �x at a time t + �tdepends only on the conditions at the position x and not on the previous history,the step at any instant being determined by a probability distribution for the jumpsin position and velocity, P (�x, �v), that depend on the properties of the medium.The same is true for the momentum, which constantly fluctuates around 〈�p〉 = 0.This independence of the impulses guarantees that the particle doesn’t drift in any


particular direction in space but instead always walks around a fixed location withever increasing distance from the origin. Over the long term, the mean positionremains zero (we’re always free to choose an origin for the coordinate system), thiscan be written as 〈�x〉 = 0 but its distance from the point of departure—the rootmean square step size,

√〈�x2〉 is not zero. As time passes the particle executes adiffusive motion due to these microscopic impulses. On average, the smallest stepsare the most probable but, every now and then, there will be the rare kick that sendsthe particle far from the origin. Keeping for a moment in a one dimensional picture,along a line we have as many negative as positive kicks—so the center doesn’tdrift—but they don’t have same magnitude and that is the key for understandingthis result—there is a distribution of the kicks. The probability for sustaining anykick is given by a normal distribution—the same one found by Maxwell for anideal gas—if there is no correlation, and the rate is determined by the diffusioncoefficient. Hence, 〈�x2〉 = 4D�t , where the diffusion term is calculated fromLangevin’s equation of motion. Although this may seem only a formalism it’s aresult of far-reaching importance. It actually provides the means to determine thenumber of molecules in a liter of gas because the diffusion rate depends on thenumber of kicks received per interval of time, which in turn depends on the gasdensity. Most important, the distribution is the same as that found by Maxwellif we look at the effect on momenta of the collisions, the individual velocitiesspread out in the same way the particles spread out in space. The circle was nowclosed.

In re-introducing impulsive forcing for local motion, we have a way to determinethe microscopic properties of matter by macroscopic observations. Each individualparticle trajectory is unique but the ensemble of trajectories isn’t. We may see anyone particle moving unpredictably, in the sense that random interactions areconstantly happening, but the motion of the collection is deterministic. The finegrained view of the world at the particle level is then replaced by the macroscopicquantities, all of which are averages over an evolving ensemble. A swarm of insectsor a flock of birds provides the best comparison. We speak of the collective (e.g.,“flock,” “swarm”) having a mean momentum and a total mass, but the structureof the group is continually changing as individuals interact with each other andtheir surrounding. There is an internal energy—each has a random as well asmean component of its motion—and any measurement can be referred to eitherstatistically on the whole or individually on the members. For this reason, the newview provided a very powerful explanation for many of the phenomena treated bythermodynamics. If the flock is in equilibrium, that is to say the individuals aremoving with respect to each other governed by some stable random process, theirmotions will be characterized by a single parameter: the temperature. If excited,they can disperse but maintain the same mean motion, similar to a heating. Butthis motion is reversible in the sense that they can reform and pass through acontinuous sequence of configurations.

Chaos

But why is anything random? It doesn’t seem to follow from the Newtonian princi-ples; in fact, it can’t. The classical equations for an adiabatic system are reversible


in time, any trajectory can be traced back as far as we want or sent as far into thefuture as we desire. For instance, Laplace had imagined a superman, a construc-tion from his treatise on celestial mechanics, who can see at any instant all theparticles in the universe and make predictions for the motion of each with abso-lute certainty. But the second law of thermodynamics seems to violate this. In ourflock analogy, we can reform the group spontaneously and there will be an infinitenumber of states—and here the phase space is actually visible—through whichthe ensemble passes. The motions are strictly reversible in the frame moving withthe mean velocity of the ensemble. Yet there are cases where this isn’t possibleand you can picture it easily by allowing the size of the flock to increase withoutbound. All birds in the group look, more or less, alike and it is only our inattentionthat prevents us from labeling each and following it. There are cases, however,where this is actually not possible and they are surprisingly simple.

The spur to the modern theory comes from the second most elementary problemof gravitational mechanics, the motion of three orbiting bodies. To be more precise,we imagine two massive bodies orbiting around a common center of mass andintroduce a third, tiny, body that orbits both. This version of the problem, calledrestricted in the sense that the third body is so small its effect on the other two canbe neglected5 and these have circular orbits. The first, analytical, solution wasfound by Lagrange. The gravitational potential is supplemented in the orbitingframe with a centrifugal term. There are then, in this frame, five equilibriumpoints, named after Lagrange. Call the two main masses M1 and M2 and themass of the third body m. Then between M1 and M2 there is a point where thegravitational acceleration vanishes. This is the L1 point. It is, however, clearlyunstable because any tiny displacement that moves m toward either mass causesit to circulate only around that mass, albeit in a very distorted orbit. There arealso two at either extreme of the principal masses, L2 and L3 where an outwarddisplacement means escape. But there are two others, perpendicular to the lineof centers, that are actually stable in the sense that a small displacement in thecorotating frame leads to an orbit around the point. These two, symmetricallylocated across the line of centers, are called L4 and L5. Because the body hasbeen displaced away from one of these points, the Coriolis acceleration (remember,this is not an inertial frame, the system as a whole rotates) produces a deflection ofthe body and since the centrifugal and centripetal forces are balanced results inthe circulatory motion. Since the frame has an orbital frequency, there is alwaysa helicity (circulation) in the system and the body moves accordingly under theaction of this force.

Henri Poincare (1854–1912) showed that the dynamics of systems near reso-nance can become chaotic. When a motion is in resonance with a driving force,when there isn’t any damping, it continues to accelerate, for instance when youalways kick at the same phase on a swing. The arc of the motion continues toincrease until the frequency is no longer independent of the amplitude and theprocess saturates. When looking at the complex dynamics of the N-body problem,not only two but, for instance, the whole Solar system, Poincare found that thereis a dense set of possible resonances, an infinite number of them, and becausethey are so different in period and so closely spaced the system rapidly becomesunpredictable because of small kicks and other perturbations. The subject was


further explored by Alexandr Lyapunov (1857–1918) in a series of studies on thedynamics of the solar system and the stability of motion near resonances.

This unpredictability also entered the microworld. In an attempt to understandentropy from a particle perspective, Boltzmann began a series of theoretical inves-tigations of the interaction between atoms that could produce the known gas laws.Boltzmann treated the transfer of momentum between pairs of colliding particlesassuming that the interactions balance. He showed the effect is to dissipate bulk,ordered motion in a sea of randomly moving particles that eventually lose theirindividual identities and, with a Maxwellian distribution, fill the available phasespace. He identified this disorder with entropy and completed the link betweenClausius’ conception of microscale motions as heat. At the core of the Boltzmannprogram lay molecular chaos, the centrality of two body interactions and the im-possibility of tracing any single particle forever. This is the same result, but forvery different reasons, that Poincare had discovered. By the beginning of the twen-tieth century this chaos had become a fundamental question in physical science,whether the stability of dynamical systems could be explored and how predictableare the motions over time. The study of chaotic behavior has developed into aseparate, rich area. The “ergodic theorem” was an outgrowth of this molecularchaos. It states that within an adiabatic system the collisions between particles,or by extension the trajectories of an ensemble of particles moving under mutualperturbations in the manner Poincare and Boltzmann had discovered, will passthrough all possible phase space configurations if given enough time and any oneof them is not privileged. The energy of the system can be defined but not theindividual motions which become too “fine grained” to be distinguishable. For theconcept of force this statistical view has an important effect: the measurementswe make on macroscopic bodies are only ensemble averages, “course grained,”so we need to compute the two body interactions and then extend them to collec-tions. For a gas this involves molecular and atomic potentials in the electrostaticinteractions and polarizations. For an ionized medium, a plasma, this includes theelectrostatic effects of free charges. For gravity, it is best seen in the collectivemotions of stars in a cluster, a beautiful example of a self-gravitating swarm.

Applications: The Atmosphere

Although the subject of speculation since Aristotle’s Meteorologia, the physicalmodeling of the atmosphere was really not begun until the nineteenth century.Two major factors contributed to this, one empirical, the other theoretical. Sincewe are imbedded in the environment, and the atmosphere is an enormous, mul-ticomponent fluid enveloping the entire planet, it is extremely difficult to have asnapshot of its state at any moment. Unlike a laboratory experiment, where thescale is so small we can determine the properties of the fluid everywhere in realtime to within the accuracy of our instruments, the atmosphere is simultaneouslysubjected to very different conditions at any moment. It’s enough to remember thatwhen it’s noon at one longitude, on the opposite side of the planet it is midnightand this means one side is being heated by direct sunlight while the other is cool-ing. Before the eighteenth century, even the coordinate system—the relative timein different locations—was largely unknown so it was impossible to coordinate


observations around the globe. Even if at one location the measurements could bemade with substantial precision, for instance the humidity, wind, pressure, andtemperature, it was almost impossible to connect these with another site on theplanet. The second, theoretical, barrier was how to combine fluid mechanics andthermodynamics. The atmosphere is not a simple liquid: it’s compressible, andthis requires a very detailed treatment of both macroscopic (fluid motion) and mi-croscopic (thermal properties, atmospheric chemistry, and the gas laws) models.Further complications come from the rotation of the planet: to solve the motion ofthis fluid you have to treat the problem as a fluid flowing over a spherical, rotatingplanet.

Three principal forces dominate its structure i.e., gravitation, pressure, andnon-inertial forces. The radial (locally vertical) acceleration of gravity must bebalanced by the pressure gradient in equilibrium. As Pascal demonstrated thepressure and density decrease with altitude since the the lower overlying weight ofthe air requires lower pressure—hence, less compression—to remain in balance.But this is a static problem. Weather, in contrast, requires a dynamical approach.What moves the air along the surface? Aristotelian reasoning served well enoughfor vertical motion, such as the formation of clouds and rain (light things rise,heavy things fall, air is a mixture, etc.), but what about wind?

To enter into the whole history of the subject would lead us too far astray.Instead, let’s concentrate on one feature of the large scale. With the gas laws,one can understand that a disturbed parcel of air will necessarily move. It hasan inertia because it’s massive. But it is also a continuum, a fluid, and thereforeresponds to both shear and stress. Across any area, a difference in local densityat the same temperature, or at different temperatures a difference in pressure,produces a reaction in the fluid. It moves with a speed that depends on the pressuredifference. Vertically this isn’t important because, overall, the atmospheric gassimply oscillates around a small perturbation and radiates waves that move atthe sound speed. But for horizontal motions, if one region is at higher pressurethan another, fluid will flow from high to low pressure. Such analogies had beenremarkably successful for electrical currents. Ironically, it was much later, almost40 years after Ohm, that the same reasoning was applied to the atmosphere.There is, however, a difference. In a circuit, or in normal small scale hydraulics,the rotation of the laboratory is irrelevant. For the atmosphere, it isn’t. For slowmotions, any pressure gradient is balanced by the Coriolis force. A region of lowpressure, for instance, should simply be a “sink” for fluid. Produced, perhaps, by alocal updraft that removes the air from a level, or by a lower density or temperature,the surrounding medium should refill the volume. The difference is that in arotating frame no simple radial motion is possible, the Coriolis acceleration deflectsthe flow. Thus, a pressure gradient amplifies circulation, this is stated symbolicallyas gradP = 2ρ × v where is the angular speed of the frame, ρ is the densityas usual, and v is the linear velocity. Since the rotation is at some angle to thelocal plane (except at the equator where it is parallel), the velocity is immediatelyobtained for a steady flow. This is called the geostropic approximation. For theatmosphere, pressure and temperature gradients are the dominant accelerationsand, for the Earth, the frame rotates so slowly the centrifugal acceleration is


negligible. Without an understanding of the steady state dynamics in responseto these forces, weather phenomena will appear mysterious and often counter-intuitive.

But notice that the motion is inevitably vortical. The importance of this for theinterplay between the dynamical and thermal state of the atmosphere was firstunderstood by Vilhelm Bjerknes at the end of the nineteenth century. Introducedto fluid mechanics by Helmholtz, he realized that the geostropic approach could beturned around, permitting a study of the forces driving the motions based only onthe observed velocities. The spur was the circulation (vorticity) conservation theo-rem, discovered almost simultaneously in 1857 by Helmholtz and Kelvin. If a fluidmoved inviscidly it conserves its circulation in the absence of frictional forces.Smoke rings and contrails are simple examples of the phenomenon, as a vortexslows down it expands. Conversely, a converging flow amplifies the vorticity andproduces, according to our discussion, a strong pressure deficit relative to the back-ground fluid. This is the well-known bathtub vortex but, as the dissipative action ofever-present viscosity—no matter how small—ultimately brings even this circula-tion to rest. Vilhelm Bjerknes, one the founders of modern meteorology, had studiedwith Helmholtz in Berlin and realized the dramatic, global consequences for vor-ticity in the control of cyclonic storms. His applications of dynamics to predictionof fluid motions in the atmosphere, including the fundamental notion of pressureand temperature fronts, became the basis of geophysical fluid dynamics. Now boththe weather and the oceans were firmly in the domain of physical investigations.

Atmospheric motions are also governed by buoyancy. For liquids this hadbeen extensively studied during the second half of the nineteenth century. Inyet another application of Archimedian principles, a locally heated compressiblefluid is buoyant relative to its surroundings. But unlike a floating body whoseproperties don’t change with location, a moving parcel of air expands and coolsif it moves into a region of lower pressure (as it will if it rises from the surfaceof the Earth). Depending on the rate of energy transfer between the interior andexterior of the parcel, it may be brought to rest by either loss of buoyancy (coolingrelative to the background) or friction. If the background temperature gradientis sufficiently steep, the parcel will continue to rise and even if it expands willremain less dense than its surroundings. The statement that the background has anadiabatic gradient, a central concept in atmospheric physics, represents the limit ofstable motion. Now combined with vorticity, this buoyancy also transports angularmomentum and is subject to accelerations independent of the gravitation, hencemaking possible substantial vortex stretching (convergence) and amplification(pressure drops). Many of these phenomena had been studied in the laboratoryand in problems studied by hydraulic engineers. They illustrate the enormouspower and scope of the merger of thermal and mechanical principles by the endof the 1800s.

Applications: Stars as Cosmic Thermal Engines

The beginnings of an astrophysics came with the merger of thermodynamics andgravitation. The development of thermodynamics and its application to cosmic


bodies, other than dealing wit their motions, produced important results ofcosmological significance. The most dramatic, perhaps, was the determinationof the age of the Sun. In 1857, Kelvin and Helmholtz independently realized thata hot self-gravitating mass cannot stay hot forever. Since it is compressible, if thelosses are slow enough that it doesn’t collapse from the loss of pressure support,it must contract in a finite time because it has a finite amount of internal energyto lose. In contrast, for the Earth, this cooling can occur without any contraction:it is already solid the internal and external temperatures merely come to a bal-ance. Since the surface is in contact with free space, the internal temperaturesteadily decreases and the rate of heat loss depends only on time. Assuming theinitial store of heat comes only from the formation of the planet by gravitationalcontraction from some initial extended mass, perhaps an interstellar cloud, therate of energy loss through the surface depends only on the temperature gradientand thus only on the internal temperature now. Fourier had described the coolinglaw for such a case, when the heat diffuses through the body to be radiated at thesurface, and Kelvin was especially familiar with this work (having introduced themethods developed by Fourier to English mathematicians in the 1820s). However,a problem had already appeared since the energy flux measured on Earth seemedto be large enough to yield a relatively short timescale, hence age, since the for-mation of the planet. But the luminosity of the Sun was already well enough knownthat it seemed reasonable to apply similar reasoning to that body. The argumentis this.

For a self-gravitating mass, by the virial theorem of Clausius we know thetotal energy must be a fraction of the binding energy, which is one half of thegravitational potential energy. Since the body loses energy, this potential energymuch continue to decrease, and if the mass remains constant this requires acontraction. But this means the internal temperature actually continues to risesince, for an adiabatic gas the pressure and temperature increase together oncompression. Thus the rate of radiative loss also increases and the contractioncontinues. Knowing only the rate of energy loss, L, since the total availableenergy is of order GM 2/R, where M is the mass and R is the current radius ofthe Sun, the timescale, tK H —the Kelvin-Helmholtz timescale—is approximatelytK H ∼ (GM 2)/(RL) which is about 10 million years. This was long enough tobe comfortable with human history but not with other, accumulating evidencethat geological and biological evolution requires at least a factor of 100 timesthis interval. Another problem came from the measurement of the solar radius. Itwasn’t changing quickly enough. This can be done in two ways. The difficult one isto directly measure the size of the Sun over a relatively short period, years, to seeif it is contracting. The other requires understanding the mechanics of the lunarorbit over a long time—you can use eclipse timing and duration to say if the Sunhas changed its radius over millennia. Both yielded negative results. There wasnothing fundamentally wrong with the calculation or the application of either theprincipals of hydrostatic balance (implying slow contraction) or thermal balance(the amount of energy lost is slow compared with the available store). The basicscenario was wrong. Both studies assumed no additional energy source for theluminosity. We know now that the energy for both the Earth and Sun are supplied


by nuclear processes, not either simple diffusive loss of an initial quantity or thegravitational potential alone.

Nonetheless, the remarkable consequence of this application of mechanics tothe stars had a dramatic effect. For the first time the irreversibility of thermalprocesses and the idea of force had produced a timescale. In an extension ofthis work, the instantaneous structure of the Sun was solved by Homer Lane in1867, and later extended by Robert Emden in 1903, to encompass the generalgas laws that an opaque body would obey. For a sphere, since the gravitationalforce depends only on the interior mass at any radius, you know the pressuregradient required to maintain mechanical balance throughout the interior pro-vided you have a relation for the internal pressure as a function of density. Thenthe radial dependence of compression, the local density, is derivable. The onlything this depends on is the central density but the degree of concentration, theratio of the central to average density, is given by the equation of state (the com-pressibility) alone. Thus measuring only the current mass (from the planetaryorbits) and radius (knowing the solar distance) you can determine the centraldensity (and therefore the central temperature). The resulting model, called theLane-Emden equation, served through the 1930s to provide the basic model forstellar structure. A modification came from including the effects of radiation pres-sure, following the developments in electromagnetic theory, as a limiting equationof state for the hottest stars. But the general picture was complete with onlythese fundamental ideas of force and heat. With this, the science of astrophysicsbegan.

NOTES

1. This image would much later in the century be put to extensive use by Stanley Jevonsin his treatises on economy. The thermodynamic model spurred much of the early workon theoretical economics during the last third of the nineteenth century and well into thetwentieth.

2. It would take us too far from our main story to describe the developments in chemistryduring the following century but one thing is very important at this intersection of the twolines of development of chemistry and physics. These substances were soon found to havespecific modes of interaction, and displayed specific relationships among themselves suchas weight and affinities to other substances.

3. This is what we mean by independent probabilities. Supposing two quantities areindependent of each other, call them X2 and X2 and each has a chance – a probabilityP (x) – of having some value x. The joint chance of having a value simultaneously ofX1 = a and X2 = b is then P (aandb) = P (a)P (b). It’s the same for the distribution ofindependent values of the different components of the momentum; in fact, the distributionfunction is the same as the chance that in an interval of momentum dp you will measurea value p.

4. A degree of freedom is an independent mode of motion. Each is a way to repartitionthe internal and kinetic energies and can be included separately in the distributionfunction. Adding new modes of motion—new degrees of freedom—is possible if theparticles have structure instead of being ideal and pointlike. Imagining a diatomic moleculeas a harmonic oscillator that can execute one vibrational and two rotational modes increasesthe specific heats—two are defined, one at constant pressure (Cp), the other at constant


volume (Cv), different for the two types of thermodynamic transformations—and makestheir ratio, γ = Cp/Cv , decrease.

5. Although it is common practice to call this third body “massless,” it clearly can’tbe. Instead we mean the center of mass is between the two principal components and thethird, which must have some mass, orbits around them.

7

FIELDS AS EVERYTHING

By no endeavor can magnet ever attract a silver churn.—W. S. Gilbert, Patience

Gravitation introduced the idea of a field of force into physics, spanning thevast distances between the planets and structuring the stars. But electricity andmagnetism were compellingly different and required a new way of thinking aboutforces. They seemed at first to require a fundamentally different treatment becausethey violate the notions derived from experiences with gravity. A recurring themethrough this chapter will be that while gravity derives from a single propertyof matter, mass, and is only attractive, electric and magnetic forces depend onthe composition and state of the material, are independent of the mass, can alsorepel, and can even overwhelm the effects of gravitation. Unlike gravity, they canbe switched on and off. They produce motion but not in all bodies, seeming tocontradict the third law of motion. And strangest of all, these forces seem to belinked to each other and not to gravity.

ELECTROSTATICS AND MAGNETOSTATICS

Imagine dividing a single bar magnet into two identical pieces so they have thesame strengths at their poles and also the same weight. Take one and suspend itabove the other with identical poles facing each other. Note the distance at whichthe two reach balance. Then increase the weight of the suspended body and notehow the distance between the two decreases. Since we can use the weight as asurrogate for force, as we’ve been doing all along, this yields a force law. Evenbefore the Newtonian treatment of gravity, this was enough to show that a repulsiveforce can have an simple, universal dependence on distance. The reverse is alsosimply found by taking opposite poles and reversing the position of the weights. Theattractive force had provided Kepler with a physical explanation for the binding of


the planets to the Sun, and it was obvious also from the orientation of the compassneedle. This second phenomenon was modeled, although not explained, by Gilbertat the beginning of the seventeenth century (and exploited by Halley at the endof the same century in his magnetic map of the world) using a spherical lodestoneto represent the earth and a slim filing to serve as a compass. With this, he wasable to account for the navigators’ results: if the Earth is a magnet, the compasstraces its influence. Unlike gravity, however, magnetism isn’t universal in thesense that not everything is equally affected and there are some substances thatdefy magnetic attraction. This non-universality was mimicked by electricity. Somesubstances could easily produce sparking and other electrical phenomena, whileothers yielded no effects although the phenomenon was far more general thanmagnetism. Electricity seems, in common experience, to come in two forms. Oneis the ability of a substance to retain charge and produce electrical phenomena,sparking being the most obvious effects and the one that first excited curiosity.Unlike magnetism, it can stored in the substance, can be exchanged by contact,and can be lost in time. Like magnetism, it can both attract and repel dependingon the substance.

The first step was understanding the nature of this force. Since it has quantity—for instance a substance can become more charged if it is rubbed for a longer time—the “electrical fire” seemed to be something exchangeable. The phenomenon sug-gests an explanation. The amount of “electricity” and an object’s ability to retain itdepends on the the substance of which the body is composed. Unlike gravity, thisisn’t something intrinsic to all mass. The model that comes to mind is, therefore,a fluid, and Benjamin Franklin (1706–1790) proposed just such an explanation.Considering a single fluid, a surplus can be called positive, while a deficit is neg-ative. Obviously, in equilibrium these cancel and produce neutrality. This singlefluid picture thus explains the attraction of bodies as an attempt to neutralizeby the exchange of the electrical fluid. Franklin’s famous kite experiment was ademonstration of his theory, as well as of the electrical nature of lightning, and nota discovery. The experiment was carefully prepared according to Franklin’s notionof the fluid exchange. In 1757, John Symmer, a member of the Royal Society, inone of the most amusing and profound examples of the application of a completelyordinary observation to a deep question, used the discharge of his woolen socksunder different ambient conditions to argue in favor of a two fluid hypothesis,although maintaining the terms positive and negative as Franklin had used them.But without a way to quantify charge or calibrate some standards the study ofelectricity could not be put on the same footing as gravity. This required a way tolink electrical and mechanical effects.

Inspired by the force law for gravitation, Coulomb set out to determine the lawgoverning the interaction of charges. He faced a difficult problem in measuring theforce between two charged bodies. Repulsion is comparatively easy because thereis no restraint on the charged objects. They can’t touch. So for this measurementhe could use the torsion balance. Two small spheres were identically charged,using a pin to transfer charge from a Leyden jar to a pair of pith balls. These weremounted on the arm of a torsion balance for which Coulomb had already performedthe appropriate calibrations. Thus the static twist of the cord of the balance gavethe measure of the force. For the opposite case, attraction, he charged a metallic

Fields as Everything 139

sphere and used induced charge on a grounded sphere to produce the oppositesign. The spheres had, however, to be kept separated so an alternate meanshad to be found for measuring the force. Coulomb had the idea of performinga dynamical measurement. Perturbing the equilibrium position, maintained bytension, he determined the oscillation frequency, ω, of the horizontal pendulum.The frequency scales with a force F as F ∼ ω2. If the force between the attractivespheres scales as the inverse square of the distance, as he had found for repulsivecharges, then the time of oscillation �t should scale as �t ∼ d . The same typesof measurements were done for magnetics with the same result: the force betweentwo objects either electrified or magnetized varies as the inverse square of theseparation to very high accuracy (about 2% in his measurements). This was thetechnique later used by Cavendish to measure the gravitational constant but it iseven more striking with electricity because this force was not part of the originalmechanical worldview.

ELECTRICAL POTENTIALS

When we discussed Green’s theory of potentials we focused on gravitation. But theroute by which he arrived at his construct was actually the study of electricity andcharge. He worked within a fluid conceptualization of the force, an atmospherethat surrounds and permeates a conductor. This electrical fluid can change densitydepending on the properties of the medium but has certain boundary conditions.Just as there must be pressure balance across an equipotential surface for a fluid,there must also be equilibrium between the electrical fluid within and outsidea conducting medium. His principal mathematical results were reached withelectricity in mind. He applied the condition that the fluid density, or potentialfunction, may be discontinuous across the boundary of a conductor. He also statedthat the component of the force perpendicular to the surface may be discontinuousbut this is balanced by the development of a surface charge. The potential functionthen depends on the continuous distribution of the charge with each volumeelement contributing to the potential weighted by its distance. Thus Green wasable to recover Poisson’s result and also match the solutions inside and outsidea conductor. For gravitation hollow spheres are irrelevant but for electrical andmagnetic phenomena these are shells that store surface charge. It was thereforepossible to compute the capacity of any shape to sustain a given potential differenceand, therefore, explain the properties of condensers.

FROM ELECTROMAGNETISM TO ELECTRODYNAMICS

Electrical and magnetic phenomena had always been studied separately althoughthey share many features. The most striking is polarity, an obvious term for a mag-net. Furthermore, until the start of the nineteenth century, magnetic phenomenahad been mainly investigated on their own using permanent magnets. When, in1820, Hans Christian Oersted (1777–1851) discovered the effect of a circuit on anearby compass needle it was not entirely by chance. He had in mind the polar-ization of the battery and the possibility that the fluid, coursing through the wire,might exert a force on a polar body. His discovery had particular drama becausethe actual first observation of the action of the circuit occurred during a lecture


demonstration in which he failed at first to see any deflection when a compassneedle was placed perpendicular to the wire. But when the needle was placednearly parallel to the axis of the current, it oriented itself perpendicularly afterfirst oscillating. In the early nineteenth century, even rather powerfully chargedbatteries had a tendency to discharge quickly so the confirmation of the effect andits exploration took some time. But by the end of the year he had also discoveredthat the effect diminishes with distance from the axis and, more significantly, thatby displacing the needle around the wire as Gilbert had around the terrela, Oerstedfound that the effect was circumferential. It was as if a vortex surrounded the wireand oriented the magnet much as a flow would orient a piece of wood. Yet thiswas outside of the current and away from the source. The effect vanished whenthe battery was turned off or spent.1 The news of the discovery spread rapidlyamong the scientific societies of Europe where the demonstration was eagerlyrepeated. It arrived in Paris with Biot. One in the audience, Andre-Marie Ampere(1775–1836), was to become, in the words of a later master figure in the study ofelectromagnetism, Maxwell, “the Newton of electricity.” Within a matter of weekshis first communication extending and amplifying Oersted’s results was receivedand with it a new physical picture was formed.

Ampere’s most important result was the discovery of mutual influence of iden-tically and oppositely directed currents. Since the battery is polar, reversing thepoles changes the direction of the flow. Thus it was that Ampere discovered thatsuspended wires carrying oppositely directed currents repel while if passing in thesame direction they attract, precisely like permanent magnets and speculated thatthe magnetism of the Earth arises from some sort of current. But the magnetic fieldof the Earth had been explained by the flow of some subtle fluid that entered atone pole and exited at the other, much like Franklin’s earliest conception of elec-tricity. This showed, instead, that the magnetic might themselves harbor internalcurrents. To explore this, Ampere reasoned that a change in the way the currentis moving through the wire would change the external field and, in particular, ifthe flow were helical the resultant field would be axial and polar. Now, it seemed,the nature of this magnetic force was seated, somehow, in the electrical actionof the battery and current. The stronger the action, the stronger the created field.Ampere coined the term electrodynamics to describe the whole range of physicaleffects he observed. His approach was to imagine current elements, analogous tocharges, along the wire. Rather than explaining the origin of the orientation of thefield, he asserted it. In a similar manner to the dependence of the gravitationalforce between two bodies on the product of the masses and of the electrostatic forcebetween two charges on the product of their charges, he asserted that the magneticforce depends on the product of the two currents. But since these are perpendic-ular to the direction of the force, the description is fundamentally different thanthe radial one for the other two, static fields.

A new force had been discovered, electromagnetism. It changed how to describethe source of a force from static charges to dynamical currents. But the effectswere still described in the same way as Newtonian gravity, a sort of action at adistance. There is some quantity of a “stuff,” charge or magnetic fluence, thatremains constant in time. Once the charge, for instance, is put into a body, itinduces a force and causes motion, but it doesn’t require something moving to


create it. On the other hand, Oersted’s experiment showed that something hadbeen actively passing through the wire of the circuit to produce a magnetic effect.Stranger yet, instead of acting along the line of centers, the force caused a torqueof the compass needle and acted at right angles. The Biot-Savart law for themagnetic field produced by a current and Ampere’s law for interaction of circuitsand Coulomb’s law for the force between static charges are all similar to Newton’sgravitational law, a force varies dimensionally as the inverse square of the distancefrom the source. The essential difference is that each element of one circuit actson every element of the other at right angles and depending on the inverse cube ofthe distance, the force between two current elements depending on their product.

Another difference between gravity and the electromagnetic fields was quicklynoted: a change is induced within the body by the presence of the field, it polarizes.For a magnet, this causes the poles to torque and anti-align. For electricity, a staticcharge image develops on the surface of the affected object, an opposite charge,when the field is applied. For gravity, the differential acceleration of the tidesmight look like this since, in the orbiting frame, the accelerations are equal andoppositely directed. But in reality, in the stationary frame, the force is alwaysattractive and centripetal and it is only because of the finite size of the body thatthe force varies across it. The mass doesn’t polarize although the distortion aligns.Nothing actually changes in the interior and the reaction, while depending on therigidity of the body, is otherwise independent of its composition.

ACTION-AT-A-DISTANCE

Gravitation was somehow communicated instantaneously between masses. In thePrincipia, no explanation was proposed—it was one of the “hypotheses” Newtonsought to avoid—but there was no pressing need for such an explanation. Thedistances could be handled without difficulty because there were two featuresimplicit in his thinking that permitted this. One was that there was no specifictimescale for any dynamical processes. If the propagation of the attractive in-fluences of masses took time, that was fine because one had all the time in theworld, literally. The other was the action of God, who was always there to adjustthe workings of anything that got out of balance. This second was acceptable inthe seventeenth century way of looking at natural philosophy and, actually, was aplus for the picture because it squared nicely with the theological preconceptionof finding in the Universe the action of a watchful creator. The former was moretroubling because it used the wrong language, the sort of semi-mystical descriptionthat irked the Continental philosophers and smacked of alchemy and the occult.

For electricity it was the same despite the anomalous feature of repulsion. Butelectrical and magnetic phenomena are not only both attractive and repulsive,they are also polar. For a magnet this is obvious, in fact it’s part of the definitionof magnetism. For electricity, there were two competing views in the eighteenthcentury. Franklin’s single fluid hypothesis used “positive” and “negative” in thesense of “excess” and “deficit.” If the equilibrium state is neutral, an excess ofcharge displaces to cancel a deficit when the two are brought into contact. Aninsulator prevents this transfer while a conductor facilitates it. This squared wellwith experiments. But it also worked well in the fluid context. For a fluid column


in equilibrium under gravity, there is no net acceleration although there is aninternal pressure that counterbalances the weight of the fluid at every point. If,however, a hole is opened in the container, a jet of fluid exits and reaches thespeed it would have were it to be freely falling from the top of the column to thehole. This can be demonstrated by noting the height reached by the jet that istilted upward and also its range. Thus Torricelli’s law of pressure, combined withthe picture of an ideal fluid by Euler and Daniel Bernoulli, described the motion.The imbalance of forces at the hole, when the fluid pressure exceeds that of thesurrounding atmosphere, accelerates the fluid. There was a dispute for some timeover the nature of this “electrical fluid,” whether there is one that is in excess ordeficit, or two that are truly attractive and/or repulsive.

The first attempts to determine the force between charges was made by JosephPriestly (1733–1804) who also demonstrated that within a conductor there is nonet force. This was one of the foundations on which Green’s potential theory wasbuilt. When Coulomb performed his electrical measurements, finding that theforce between static charges depends only on the inverse square of the distance, itseemed a confirmation of the universal picture of forces and action-at-a-distancethat Newton had propounded. In fact, it was precisely because Coulomb wasdealing with a static effect that this was plausible. The measurement was madeusing a torsion balance, adopting the technique also independently proposedby Mitchell for measuring the gravitational attraction between two masses andlater adopted by Cavendish (as we will discuss shortly). In addition, Coulombdemonstrated the existence of two separate charges that, when alike, repel andotherwise attract.

When it came to electrodynamics, specifically Oersted’s and Ampere’s discov-eries, this started to present a conceptual problem. Ampere could treat the effectsof circuit elements as if they were charges, little pieces of lines in closed circuitsthat in the presence of a steady state could act for a long enough time that prop-agation effects were negligible. The Biot-Savart law and Ampere’s law were bothcouched in this language, with the magnetic field being proportional to the currentin a wire and the force between two wires varying dimensionally as the inversesquare of their separation and the product of their currents. But open circuits werea different matter. First we need to distinguish the two cases. A closed circuit issimple, for instance a loop of wire connecting the poles of a Voltic pile. A flow ofelectricity, whatever that might be, within this “pipe” creates a magnetic field andthat is the cause of the repulsion or attraction when two such circuits are broughtinto counter-alignment or alignment, respectively. An open circuit, in contrast, isa discharge, and this phenomenon also differentiates electrical and gravitationalphenomena. It is possible to collect charge—in the language of the eighteenthcentury, to “condense the electrical fluid”—for later use; the device, called aLeyden jar. Franklin demonstrated that this effect can be multiplied by combiningthe jars in series, he called it a “battery of condensers,” and showed that thestored charge is held within the insulator (later Priestly showed the glass can bereplaced by air). There is a finite quantity of charge that can be stored, dependingon the insulating substance, again unlike gravitation. The charge effect residesat the surface of the insulator, the part that is either in contact with the metallic


shield or at the gap when they are not in contact. When the Leyden jar releases itscharge it obviously takes time. This too is completely different from gravity. Sincethe source isn’t in steady state, its effects on anything in the vicinity should alsodepend on time and to require propagation of the influence. Static approximationscould not explain this effect. Several laws were proposed that included the veloc-ities and acceleration of the elements of the electrical fluid and/or the circuit butnone of these were completely specified nor epistemologically consistent with theprevailing physical theoretical picture.

Circuits as Flows

A static fluid representation had served well for potential theory. It allowed asimple mechanical image of how a charge medium could accommodate itself tothe presence of a central, electrified body. The discovery of electromagnetism, theconnection between a time dependent electric potential and the generation of amagnetic field, suggested an extension to a dynamical fluid. This was accomplishedby introducing currents. The model was remarkably successful. You’ll see shortlythat even Faraday’s force lines could be expressed mathematically as results ofcurrents generating vortical motions of an imbedding continuous medium thattransmit the electrical influence to magnetic bodies. In this way, the Biot-Savartlaw and Ampere’s treatment of sources were accommodated in a single explanatorypicture. It wasn’t only fruitful for explaining the generation of magnetic activityby a discharging pile.

When a wire connects the poles of a battery, a current flows from one site toanother. In the single component picture, this is the transfer of the “electricalfluid” from that which is in excess to the deficient side. It was clear that thiscan occur in free space, or in air. To have it pass in a closed circuit meantthat the charge was moving as if in a pipe driven by an electromotive force, theEMF. The hydrodynamic analogy of currents, sources, and sinks was effectivelyemployed by Georg Simon Ohm (1789–1854) to study the passage of currentsthrough circuits and their resistive elements. For us, this is only a sideline butwe still have reminders of this earlier era in the terminology of physics andelectrical engineering. The EMF, which is now called the potential difference orvoltage, records the confusion between force and energy that persisted through thenineteenth century. In an electrical circuit, the EMF is the source of the current,the battery or power supply, and a current flows in response. The hydraulic examplewould be water issuing from a container, the electrical analogy is a condenser,placed higher than the sink so a current flows through the tube and then using apump to raise the water again, the analog of a battery. This immediately connectsthe mechanical and electrical forces, a potential difference is needed to createa flow and maintain it. The introduction of resistances to the flow can occur inseries or parallel, you can think of this as joints in the piping or junk cloggingthe tube, thus impeding the flow, and the two laws that result are that the sum ofall currents is constant and the sum of all drops in the potential must equal thepotential difference of the source. For our discussion, this amounts to a new wayof seeing how a force is transmitted through a system, how various components act


as modifiers and amplifiers. The capacitor, resistor, and inductor were all loadsthat impeded the flow of the current. A condenser was not only, as with a Leydenjar, something with the capacity to store charge. It became a dynamical elementof a circuit, capable of both charging and discharging in time. This general fluidconception, first subjected to systematic mathematical treatment by Ohm, alsoimmediately connected the electromotive force with the energy, or potential, of thepile. The general law is that the potential difference across a circuit elements isequal to the current times the resistance. Further use of the flow analogy was madeby Gustav Robert Kirchhoff (1824–1887) in his analysis of how loads distributethe current and potential difference. Loads placed in the circuit in series, that isin immediate succession, are directly additive. This is because the total drop inthe potential across the sum of the elements is the same as between the poles ofthe pile. The same quantity of current must pass through each element, as wouldhappen in an incompressible fluid. Therefore, Ztotal = Z1 + Z2 + · · ·, where Z j

is the impedance (the generalized load). On the other hand, as with a cascade,parallel elements experience identical potential drops but pass different amountsof current. Think of the different parts of Niagara Falls: the height from whichthe river drops is the same for all channels but they pass different quantities ofwater. The potential difference is now, however, not due to gravity but electricfield that develops across each elements. Thus, for parallel loads the impedancesadd in inverse series. 1/Ztotal) = (1/Z1 + 1/Z2 + · · ·. The reigning confusion interminology between “force” and “energy” persisted, electromotive “force” (stillso-called) is actually the potential difference across the circuit.

Electrical phenomena, especially those of circuits, also displayed links tothermodynamics. Resistances become hot, wires glow, and the voltic pile getshotter as it operates. For instance, while the temperature is defined only for athermal system, the pressure and volume can be directly relate to dynamicaland geometric quantities. Since the work done by the system is related to thetemperature, T , which is related to the internal energy. This is precisely whathappens in a battery, a chemical process releases energy and, somehow, alsogenerates a potential difference that does the work of driving a current. A scent ofunification was in the air. This connection between work and mechanical reactionwas the central feature of Helmholtz’s memoir On the Conservation of Forces in1847, which I introduced when discussing frictional forces and dissipation. It wasthe first precise treatment of energy. Although it later became the standard surveyof the subject and is now considered a classic, its significance was not immediatelyappreciated. Helmholtz has serious difficulties getting it published since it washis first major contribution in physics written while still in his twenties.2 You’llnotice the title. Even at this comparatively late stage in the development ofmechanics there remained an ambiguity between the words “force” and “energy”but the memoir makes the reason obvious. Helmholtz started with the dynamicalequations and showed that for a central potential field, the work plus the vis viva,the kinetic energy, is constant. He then showed how to include friction, the lossof vis viva, in this sum by introducing the concept of internal energy, which hecould then link to heat and temperature. Perhaps the most strikingly originalpart was, however, the connection he drew between the voltic pile—current inelectrical circuits—and energy. Helmholtz proved that the rate of power—the


Figure 7.1: Hermann von Helmholtz. Image copyrightHistory of Science Collections, University of OklahomaLibraries.

rate of energy dissipation—for a resistoris related to the electrical potential driv-ing the current and the resistance, muchas viscosity in a flow slows the motion.As I’ve said, a notable phenomenon incircuits is that the components heat up.This connection between thermodynamicsand electrodynamics seemed like an ef-fect of friction, almost a viscosity of thecurrent in the circuit. His result was thatthe power = I 2 R for a current I and re-sistance R. He already knew, from thedefinition of capacitance, that the chargeis related to the potential difference andsince the charge times the EMF is the en-ergy (analogous to the mass times the po-tential), the current times the EMF is thepower (the rate of change of energy withtime). It was thus possible to extend all pre-viously developed fluid mechanical expe-rience to electrodynamic phenomena andaccount for gains and losses. The concep-tual framework was very general and in-dividual devices could be included in thesame way as Kirchhoff ’s treatment of theimpedances.

Induction and Lines of Force

Charge also induces charge; when near a neutral material a charged body canproduce an electrical response. Charge can be transferred from one material toanother by contact and stored in some and not in others. A glass rod can holdcharge and produce a polarization and conductors behave as if they have equal andopposite charges when placed near a source, charge images that are the analogs ofoptical mirrors. Such static effects were well studied by the 1830s, although notunderstood. To add the confusion, when Joseph Henry (1797–1878) and MichaelFaraday (1791–1867) discovered electromagnetic induction and self-inductionthey introduced a new wrinkle into the concept of force, that one agent in theinteraction could induce a similar state in a previously undisturbed body andeven have a back-reaction on itself. This electromotive force was different fromthat driving a closed circuit with a battery but it required a closed loop and atime variable source. Both also discovered self-inductance, the back reaction ofa time dependent circuit on itself, but it was Faraday who recognized the deeperimplications of the discoveries and profoundly altered the course of physics.

For gravitation, the presence of another mass produces a reaction but doesn’tinduce anything. The force depends only on the quantity of matter. The same istrue for an electric field, as we’ve seen, the main difference being the possibility


Figure 7.2: Michael Faraday. Image copyright History ofScience Collections, University of Oklahoma Libraries.

of static repulsion. But both forces areradial and steady, if the source remainsconstant. For electromagnetism, induc-tion is something completely different.The time dependence was the clue thatFaraday used to develop a working hy-pothesis of enormous reach: lines offorce. Placing iron filings on a sheetof paper lying on top of a permanentmagnet, the filings delineate the curv-ing lines of force that permeate spacearound the source. The same visual-ization technique, applied to a loop ofwire, revealed what in the surround-ing space what had already been tracedout by the motion of a compass needle,that the curved lines look like a vortexaround the source—the current carry-ing wire—and have the same proper-ties as those of a magnetic. The visu-alization is now so familiar, indeed it’snow a standard demonstration, it maybe hard to imagine when it was new buteveryone who sees it is entranced by thebeauty of the effect. The discovery wasannounced in 1851 by Faraday in the28th series of his Experimental Researches in Electricity and Magnetism in whichhe presented a new model for how a force is communicated by a magnetic fieldand thus established a novel and far-reaching program for the study of fields ingeneral. He grounded the description in operational terms—how do you measurethe direction and strength of the field—but to do this he needed to use the responseof a magnetic needle and, therefore, instead of calling this topological “thing” afield, he introduces the term lines of force. His description is beautifully clear andI’ll reproduce it in full here.

(3071) A Line of magnetic force may be defined as the line with is described bya very small magnetic needle, when it is moved in either direction correspondentto its length, that the needle is constantly a tangent to the line of motion; or it isthe line along with, if a transverse wire be moved in either direction, there is notendency to the formation of any current in the wire, whilst if it is moved in anyother direction there is such a tendency; or it is that line which coincides withthe direction of the magnetcrystallic axis of a crystal of bismuth, which is carriedin either direction along with it. The direction of these lines about and amongstmagnetics and electric currents, is easily represented and understood, in a generalmanner, by the ordinary use of iron filings. (3072) These lines have not merelydetermine direction, recognizable as above (3071), but because they are relatedto a polar or antithetical power, have opposite qualities or conditions in opposite


directions; these qualities which have to be distinguished and identified, are mademanifest to us, either by the position of the ends of the magnetic needle, or bythe direction of the current induced in moving the wire. (3073) The point equallyimportant in the definition of these lines is that they represent a determinate andunchanging amount of force. Though, therefore, their forms, as they exist betweentwo or more centers or sources of magnetic power, may vary very greatly, and alsothe space through which they may be traced, yet the sum of power contained in anyone section of a given portion of the lines is exactly equal to the sum of power inany other section of the same lines, however altered in form, or however convergentor divergent they may be at the second place.

Faraday’s language here is precise and geometric. Although not explicitlyexpressed in symbols, it leads to the differential equations for the lines of constantforce; in fact, the words translate virtually directly into symbols such as divergenceand line sections. As he states, these can be easily visualized by sprinkling ironfilings on a sheet of paper above a bar magnet, although Faraday doesn’t note thatthis is actually visualizing the field in three dimensions since the filings near thepoles will stand perpendicular to the paper due to the convergence of the field.You might be thinking that the idea of field lines, indeed of field in general, couldbe read into Gilbert’s description of the terrela. Indeed, it often is, but there areseveral important differences. Faraday’s experiments contained a basic ingredientnot found in the early descriptions of the environment around a magnetic: forhim, there was an intrinsic connection between the fields within and outside ofthe magnet. The effects are not due to emanations, humors, or emissions, theyare probing a field, a static space filling force. When these lines are changed atany region of space, that is when the field is altered, there are physical effects.Instead of being static entity, the field is active and it can induce time dependentreactions. It is truly a force in the sense that it can produce an acceleration.

When Faraday discovered the time dependent effect, that a change in the fieldproduces a current, he saw in this a generalization of the force line idea. If thenumber of lines of force passing through a closed circuit element, such as a wireloop, changes with time, it produces an electromotive force that induces a currentthrough the wire in the absence of a permanent driver (e.g., a battery). The fasterthe change, the greater the induced current, therefore the greater the EMF. Thiseffect could not be instantaneous because it depended on the rate of change,with time, of the flux passing through a closed area and it was the first electricalor magnetic effect that could not be produced by a static field, as Faraday alsodemonstrated. Importantly it didn’t matter whether the magnetic or the circuitmoved, thus showing that the cutting of the field lines is completely relative. Thus,with a single observation, the concept of force permanently changed. It was nolonger possible to ignore completely the propagation of the effect of the source.Inserting a soft iron rod in an electrified, current carrying coil produced a magneticfield; winding another wire around some other free part of the rod reacted withan induced current that depended on the rate of change of the driving current.This is a “transformer.” With this field model Faraday could describe a broadrange of new phenomena. For instance, if a loop of wire is moved so it cuts thesefield lines, it induces an electrical current. What’s required is to change the field


passing through the circuit in time. This can happen either because the strength ofthe field is changing, as demonstrated later by Lenz with a moving magnet passedthrough a coil that generates a current to induce an oppositely directed field, orby the relative motion of the wire and the field as in the dynamo. The applicationcame rapidly: a generator of electricity that could convert motion into current and,in turn through the action of electric fields, produce work.

The Dynamical Basis of Electromagnetic Units

How do you standardize the units of electromagnetism? Within mechanics, mass(M), length (L), and time (T) are the basic quantities and because of the force lawcan be used to define all dynamically related quantities. For instance, the unit offorce (the dyne or newton, depending on whether cgs or MKS units are employed)can be written dimensionally as M LT−2 (that is, mass times acceleration). Anystress is the force per unit area, so whether pressure or compressional stress this isM L−1T−2. Energy is M L2T−2, and so on. When we’re dealing with electrical ormagnetic fields, this isn’t so obvious. How do you define the unit of electrical poten-tial? Force is the key: the potential for a gravitational field has the same dynamicalrole for mass that electromotive force—potential difference—has for electricity. Itproduces a work and thus has the units of energy per unit charge. Since the electri-cal interaction between point masses is a force, dimensionally Q2L−2 ∼ M LT−2,where Q is the charge, producing the unit for charge M 1/2L3/2T−1. Recall we usedthe same argument for the gravitational interaction to find, using the elementaryunits, the dimensions of the gravitational constant. To extend this to any otherfield components is relatively simple. Since potential differences produce work,the units must be the same as energy so that QV ∼ M L2T−2 which, when we havedefined the potential difference V as a line integral over the electric field providesa definition of the electric field E ∼ V L−1. Further, defining capacitance as thecharge retained by a material acted on by some potential, these units are extendedto indirect properties of charged systems. The point here is that without the thirdlaw, it is impossible to consistently use elementary dynamical principles to relatethe causes of the phenomena and their magnitudes (the fields) to their effects (e.g.,currents in circuits). For magnetism we can continue along the same lines usingthe inductive law. The galvanometer and ammeter depend on the electromagneticinteraction between coils and fields, through currents. Static magnetic fields aremore easily arranged using naturally occurring materials (e.g., lodestones) whoseeffects can be measured. But for standards, the units for the fields depend onthe electromagnetic principles. Because only the static electric field is defineddynamically, without the induction law, hence the dynamo, it isn’t possible toobtain a set of dimensioned quantities to represent the magnetic field.3

MAXWELL: THE UNIFIED THEORY

Electrodynamical effects, especially induction, cried out for a unification. Theinter-convertibility of the actions of the two fields, depended on both space and timeand, although was fundamentally different from gravitation, nevertheless could beunderstood mechanically with the new conception of field that Faraday introduced.


Figure 7.3: James Clerk Maxwell. Image copy-right History of Science Collections, Universityof Oklahoma Libraries.

At this stage, with the field concept still beingdebated, we come to the work of James ClerkMaxwell, who has already appeared as one of thefounders of statistical mechanics. His success-ful combination of electricity and magnetismwithin a single dynamical theory representsthe first unification of the fundamental physi-cal forces and established the research programthat is still pursued today. Although Newtonand later researchers extended mechanics, thetwo developments—researches in electrical andmagnetic phenomena and those of mechanicalsystems—remained disjoint. Helmholtz, We-ber, and Kelvin, in particular, contributed manyelements to the developing picture. But the uni-fication was accomplished by Maxwell alone,although the clarification of the nascent the-ory was necessarily carried on by others thanits author, who died at age 48 of abdominalcancer.

Maxwell’s innovation came from taking thefields as properties of the medium. The field outside a dielectric, for instancethe field in free space between the plates of a condenser, is the electric fieldE. A dielectric, insulating material inserted between the plates can increase thecharge storage capacity of the condenser. This is because it polarizes, developinga surface charge produced by E and the field inside the dielectric is D. BecauseMaxwell viewed this as a response to the stressing of the dielectric by E he calledit the displacement; it was an electrical analogy to the strain of an elastic medium.By a further analogy, the external field produced by a magnetic is B and thatinside a paramagnetic substance is H. The electrical polarization, P, alters theelectric field within the dielectric, the magnetic polarization M does the same forthe magnetic field. This led to the pair of definitions, D = E + P and H = B +M. Both P and M

¯are induced and depend on the external field strengths. The

field in the medium, if stationary, is produced by the polarization of the charge,this was already known from the behavior of dielectric materials that reacted toan external field by displaying a surface charge. The change in the field betweenthe interior and exterior, as envisioned first by Green, could now account forthe surface charge in any medium with a finite resistivity. The field within themedium is then given by D and the internal field is reduced by the production ofa polarization P which produces a surface charge σ . Thus the difference betweenthe internal and external displacement current is the origin of the surface fieldand the divergence of the electric field is given by the presence of a net chargedensity. Maxwell could then write two proportions that depend on the compositionof the material, D = εE and B = µH where ε is the dielectric constant and µ

is the magnetic permeability. Neither of these was known in an absolute sensebut could be scaled to the field in “free space,” so the limiting values of the twoconstants are ε0 and µ0, respectively.


But what about dynamics? Maxwell knew from Coulomb and Cavendish4 thata vacuum field produced by a single point charge that is, it produces a flux whoseintensity falls, as for gravity, as the inverse square of the distance. But he also knewthat within a medium, the divergence of the field strength is produced by a changein the density of the charge. This was Poisson’s, Green’s, and Gauss’ result. Anychange in the charge distribution with time is equivalent to a current, but by thecontinuity condition it must be balanced by an influx or outflow of current. Sincehe could write an equation for the change in the field in time using the Poissonequation to link the divergence of the internal electric field, D to the density, therate of change with time of the density was the same as the change in time of thedivergence of the field. Maxwell then distinguished two types of currents. One isthe free current, J, a charged fluid for which an imbalance of the rates of inflowand outflow through some volume produces a change within the volume of thedensity of free charge. This is the continuity condition and is precisely analogousto the flow of a compressible fluid. This free current requires a driving agent, theelectromotive force or potential difference across a circuit. But continuing theanalogy with an elastic medium, Maxwell introduced the displacement current. Hereasoned that there is some kind of strain on the material produced by the stressof the applied field so if the field changes in time so should the displacement.The orientation of the electrical units changes but, being bound to each otherand rigidly situated in the material, the charges cannot move freely. Instead, theyreduce the internal field while accumulating at—or toward—the surface of thebody, producing a time dependent reaction and a variable polarization when anychanges occur in the imposed electric field. The effect had already been quantifiedby Faraday and Mossotti had linked it to the index of refraction of a medium. Thesame reasoning applied to the magnetic field. We have a link between a fluidand elastic picture for the reaction of any body. The mass, you notice, doesn’tmatter.5

The electric field can be obtained from a potential, which is a scalar quan-tity, whose directional changes in space—its gradient—gives both the strengthand direction of E. The magnetic field, on the other hand, is a polar quantity(that is, after all, one of its main properties, it has a handedness) so it must bedue to a different kind of potential function. To find it, there are several clues.One is Ampere’s law. Since a current is required to produce a magnetic field,the potential must be a vector (remember, this is the current density J and notsimply the number of charges per unit time). Certainly some sort of spatial deriva-tive is needed as well to produce a field but, in this case, it must have polarity.The mathematical representation was known from fluids, it’s the same one re-quired to describe vorticity and Maxwell therefore adopted a circulatory potentialfunction A. The two fields must be unified, eventually, because the current con-nects these through the induction law since the motion of a current through afield induces an electrical potential difference. Then the current serves as thesource term for the vector field just as the static charge is the source of theelectric field. The Biot-Savart law states that a current is necessary to produce amagnetic field in space and, following Oersted’s experiment, that this field actsbetween circuits perpendicular to their individual elements. But at this stage


the treatment becomes truly novel: even without sources, a time dependent strainin the medium acts like a current. For Maxwell, this meant that the medium—the ether—if subjected to some forcing, would propagate that disturbance byitself.

To sum up, Maxwell required four fields, instead of two, with different sourcesand different constraints. The internal fields are the displacement and the magneticflux. The external fields are the electric field strength, and the magnetic fieldstrength. The connection between these depends on the properties of the medium.For the magnetic field, this is the susceptibility while for the electric field thisdepends on the dielectric constant. Both of these are measurable and linked tothe energy of the field contained in an arbitrary volume, V, which he found isthe integral Energy = 1

2

∫[E · D + H · B]dV . The first term was already known

from Faraday’s investigations, that the energy stored in a condenser is the workrequired to move the charge by some displacement. If we take the displacement,D, literally, then the work in a volume element is the force density applied alongthe direction of displacement, E · D and similarly for the magnetic field.

THE ETHER

Newton was able to avoid openly describing the origin of gravity, leaving it asideas a metaphysical problem (although not without much private speculation on thetopic). It was enough that the force was shown to be identical for all matter andwith this to explain a vast swath of physical phenomena. Even when the fieldconstruct was introduced along with its representation by a potential from whichforces arise, it was still possible to avoid asking questions about the materialproperties of this medium. You could have action at a distance without detailinghow it happens. There was no such luxury when it came to electromagnetism. Thediscovery of induction showed that time is fundamental to its action. The flow ofcharges, a current, produces a magnetic field and the rate of change of a magneticfield produces a current. Optical phenomena come from the transmission of atime dependent field between a source and receptor and this seems to require amedium. If the currents behave like fluids, then waves in fields should behavelike waves in fluids with tension, inertia, and elastic properties. It seemed a returnof a discontinued item from the catalog of models, a space filling substance thattransmits force. This time it wasn’t something as fuzzy as the universal Cartesiansubstance but it bore striking resemblances. This hypothetical electromagneticallyactive medium required vortices for magnetic currents and the ability to supportwave transmission so some sort of elasticity. By the boundary conditions forelectric and magnetic substances, the limiting condition for a space containingno free charges—a vacuum from the point of view of matter—still required somedisplacement current. Just as the index of refraction for light being defined asunity in the limit of free space, the dielectric constant and permeability of theether were constants “in the limit”; a vacuum state is simply one that lacks freecharges but can otherwise be completely filled. The same was true for gravitationbut the Newtonian concept required an impulsive transmission of the force andotherwise eschewed any supporting medium as being unnecessary.


To explain induction, Maxwell translated Faraday’s notion of force lines into alocal setting, stating that the change in the magnetic flux passing through a circuitinduces a potential difference. The integral expression of this law has a strikingresemblance to vortex motion, a point not lost on either Kelvin or Maxwell. If wedefine the potential difference as a sum over parts of a circuit element throughwhose area passes a time variable magnetic flux, then the electromotive force isgiven by the change in time of the magnetic flux passing through a closed circuit orloop. Kelvin and Helmholtz had used a similar appearing definition for circulation(arriving at the theorem that in inviscid fluids this quantity is constant). The vortexanalogy was strengthened by the condition that the magnetic flux cannot diverge.Only circulation is possible for the “fluid” supporting the field, since a vortexmust either close on itself or start and end only at the boundaries of the medium.In a boundless medium, as the ether was thought to be, the second option makesno sense so the field must be produced by dipoles or even multipoles of higherorder. The system is now closed. One set of equations describes the conservationconstraints, those are the divergence conditions. The other pair are the evolutionequations for the fields as they couple. A potential difference is created betweenany two points in space merely by changing the magnetic flux passing throughany area. Consequently, a force depends only on a local variation of a field thatmay have a very distant origin. The other is that even absent free current, and thiswas Maxwell’s condition for a vacuum state, the magnetic field could be changedlocally by the straining of the ether. To be more precise, if the ether is space-filling there is no real vacuum—empty volume—anywhere. But it will seem to beempty in the sense that no matter exists to support, by its conduction of a freecurrent, a local source for the field. In the limit that the free current vanishes, twolinearly coupled fields persist, (E, B) that because of the divergenceless propertyof magnetism propagate as a transverse vibration of the ether. This solution, whichdepends on the material properties of the ether in Maxwell’s theory, combine toform a wave equation. Thus the unification of the forces governing the motion ofcharged particles produces a vibratory mode of energy transport that resembleslight. It was this result, so sweeping in its conclusion, that was the fundamentalbreakthrough of the theory. If only currents within the fluid were allowed, no waveswould result since because of the shearing of the electric field, the divergence ofthe free current vanishes and the density of charges decouples from the problem.But because of the strain term for the displacement current, there is always a timedependence produced by the imposition of a time variable field and the systemcloses without invoking the electrostatic limit. Even polarization of light could beexplained because the vibrations occur in a plane perpendicular to the directionof propagation of the wavefront. Circular polarization was just a change of phaseof the two electric field components.

You can see how this enters from the equations for the two field strengthsE and H fields in the absence of free charges. The constraint equations for themagnetic flux and the displacement, their divergences, both vanish (this impliesthat the medium is incompressible) and the time-dependent equations for the twofields combine in a single equation. The argument proceeds this way. The spatialvariation of the magnetic field, even when no free current is present, is producedby the time dependent differential strain—the displacement current—while the


spatial changes of the electric field are produced by time variations of the magneticflux (or induction). The two resulting equations are linear and the two componentsof the electromagnetic field don’t interfere with each other. The time and spacedependent equation that resulted for each field separately is a wave equation, themathematics of which had been studied for over a century since D’Alembert andEuler. Maxwell immediately recognized it and, further, identified the propagationspeed with the product of the two material constants that are the analogs ofmechanical strains, ε and µ. This dynamical field now propagates through theether with a constant speed that depends only on the electromagnetic properties ofthe medium. While forces, as such, are not mentioned here, we are clearly dealingwith a mechanical problem in which a continuous medium is reacting by a strainto an imposed, time dependent stress. This field is not passive, and the mediumthrough which it is transmitted isn’t either.

This was the most important prediction of Maxwell’s unification, the featurethat distinguished it from any other theory for electromagnetic phenomena: forfields that vary in time there must be electromagnetic waves. The near match ofthe value for ε0µ0 with c2, the speed of light squared, was strikingly successfulin itself. But the more significant predictions of the theory were that there mustbe polarized electromagnetic waves that move at the same speed as light. Theexperimental verification of this prediction was accomplished by Heinrich Hertz(1857–1894), a student of Helmholtz. Remember that visible effects, sparks, werewell known to accompany discharges under the right conditions so there wasan obvious connection between light and electrical phenomena. This was not soclear for magnetic phenomena but we’ll return to that point in a moment. Sincethe frequency of the electromagnetic wave depends only on that of the source,which must have a change in time to produce emission, Hertz’s idea was to usea simple wire dipole connected to a galvanometer. By discharging a spark onone side of a room he measured the jump in the galvanometer. From the timedifference, which is a much harder measurement, he knew the speed of the effectregistered by the passive detector. In 1892 he completed the series of experimentsthat conclusively demonstrated the existence of electromagnetic waves, findingthat the speed of propagation closely matched the velocity predicted by Maxwell’stheory and more important, the already known speed of light. This was not action-at-a-distance but the transmission in time of a disturbance. The technologicaldevelopment of radio was the result of increasing power in the transmitter andimproved sensitivity by using resonant detectors, crystals. By the beginning of thetwentieth century transatlantic signaling had been achieved by Marconi and radiohad arrived.

Radiation Pressure

There was, however, a special feature of this transmission. A boat can be accel-erated by a wave when it is hit and the wave scatters. This momentum transferdepends on the energy in the wave and the mass of the boat. If we think instead ofa charge being hit by an electromagnetic wave the same thing will happen. Sincethere is a flux of energy, there is also one of momentum because the propagationis directional. Analogously, a consequence of the electromagnetic unification was


the prediction of a new force due to the incidence of light on matter. You mightthink this was already built into the Newtonian construction for light, particlesmoving at a speed depending on the medium that carry momentum, but this wasnever explicitly explored amid the flurry of applications to optical phenomenaand was largely overshadowed by the rise of wave theory after the start of thenineteenth century. Once a comprehensive physical/mathematical derivation hadbeen achieved to support the wave picture, it was clear that the flux of energytransported by the waves can impart an impulse. The decisive discovery was byJohn Henry Poynting (1852–1914), after the publication of the Treatise on Elec-tricity and Magnetism. Since in a vacuum the wave propagates at a fixed speedand the vibrations are in the plane perpendicular to the direction of propagation,Poynting was able to show that the flux of energy depends on the product of theelectric and magnetic field amplitudes and is in a direction perpendicular to theplane of vibration, and the momentum imparted to a particle on which this fieldis incident is the flux divided by the speed of propagation. Now there was, finally,a direct connection with forces in the Newtonian sense, although not by steadyfields. In this case, a time dependent field will move the particle in two senses,around the direction in which the wave is propagating and along it depending onthe inertia and charge of the particle. We have now an impulse that depends on thecross section of the particle to the interaction with this wave. In other words, lightexerts a pressure.6 We now know that the Poynting flux does, indeed, acceleratebodies, including satellites and even dust grains orbiting the Sun, an effect firstdiscussed by Poynting himself and later elaborated in the relativistic context byH. P. Robertson almost 50 years later.

Another effect of radiation pressure connects electromagnetic theory to thermo-dynamics; the link was realized by Boltzmann in the 1880s. Imagine a completelyenclosed, opaque container containing only radiation and perfectly absorbing andemitting walls. This is a blackbody, an idealization of a body that both absorbs alllight incident on its surface and then reradiates it as it heats with a distributionthat depends only on temperature until, when equilibrium is reached betweenthe rates of emission and absorption, the radiation (whatever its original spec-trum) has precisely the same energy density, ε as the walls. Now since this is anisotropic process, the radiation energy density is only a function of temperature,T , and the total energy is simply the energy density times the volume of the box,E = εV . Being adiabatic, the process occurs at constant entropy and thereforethe work done by the radiation is precisely balanced by the change in the heatcontent of the walls. Thus the radiation pressure, which is normal to the walls,is Prad = 1

3ε(T ). Boltzmann showed, by a purely thermodynamic argument, thatε(T ) = constant × T 4; this agreed with the already known empirical law from Ste-fan’s studies of thermal radiation. It followed, therefore, that the radiation wouldalso exert a pressure on the medium whose ratio to the gas pressure increases asPgas/Prad = εgas/εrad ∼ ρ/T 3 where ρ is the mass density. Returning now to ourdiscussion of stellar structure, this had an additional application—the structureof very luminous stars may be dominated by radiation pressure even when thestars are obviously composed of gas. This last point was exploited by Arthur S.Eddington in the 1920s and 1930s to extend the solar model to the most massive,therefore hottest, stars. So what started out as a unification of two forces of nature


becomes a means for understanding the structure of the most massive bodies thatemit the radiation.

Acoustics

The development of acoustic theory occurred parallel to electromagnetic theory asa sort of proxy study. An acoustic wave is a pressure disturbance that propagates asa wave in a compressible medium, e.g., air. If the gas is in contact with a vibratingsurface, for instance a plate, it responds with successive rarefactions and com-pressions to the fluctuations in the driving surface. Unlike light, these don’t moveat a constant speed unless the medium is uniform, and unlike electromagnetismthe waves can be driven by a monopole (that is, a pulsating sphere that varies onlyin its radius), but all of the physics is there. The waves damp because of molecularmotion, the Navier-Stokes treatment of the viscosity dissipates energy as heat, butthe equations of motion in an ideal medium are identical with those of an electro-magnetic wave. It should, therefore, not be surprising that most of the founders ofelectromagnetic theory, at some time in their scientific lives, studied fluid and/orgas wave phenomena. Lord Rayleigh (John William Strutt, 1842–1919), in partic-ular, was especially concerned with this, founding the modern science of acousticswith his Theory of Sound and hundreds of separate studies of topics from waveproduction to radiation patterns for musical instruments. But it’s also about forcesand how the boundary conditions change the solutions, describing how differentdevices, or subsystems, couple together. Sound is a pressure disturbance andtherefore the force depends on the area for a given amplitude. Fluctuations indensity, or pressure, can be in resonance if their wavelength is a rational multipleof the size or length of a cube. In EM theory, this is the same as a waveguides. Ifwe send a wave down a conducting, or dielectric, channel, there will be resonantmodes that depend on the shape of the waveguides. These depend also on thedielectric constant and the magnetic permeability of the walls. Thus, a horn thatproduces a set of overtone pulsations of air has an analogy with a waveguides in aconical or horn shape for an electromagnetic wave. If we assume, as Maxwell andhis contemporaries did, the ether is a continuous elastic medium, it is possibleto analogously treat the propagation of sound and the electromagnetic field. Eventhe concepts of wave pressure, of wave momentum transfer, and of harmonics andnonlinear scattering (when the waves interact, strong waves will generate sumand difference frequencies, a ophenomenon discovered by Helmholtz and now thebasis of much of nonlinear optics), and many analogous effects find their corre-spondence in acoustics. In effect, it became a model system for Rayleigh and hiscontemporaries and later, especially during the effort to develop radar during theWorld War II (at the Radiation Lab at MIT, the team led by Philip Morse andJulian Schwinger) the analogy became a powerful tool for predicting optimal formsfor the transmission lines of the waves and developing resonant amplifiers anddetectors.

Dynamics and Electron Theory

Recalling now Faraday’s result for induction by dynamo action, we know thatit doesn’t matter if the source of the field or the receiver of the action is in


motion. This is the starting point for H. A. Lorentz’s (1853–1928) extension of theprinciple to the motion of charged particles. A current is deflected by a magneticfield, this is the Hall effect that was discovered by Edwin Hall (1855–1938) andstudied extensively by Henry Rowland (1848–1901). Hall found that a magneticfield placed across a conductor produced a potential difference transverse to theelectrodes of the flow of the current. It is another manifestation of induction. Thevariation in resistance identified the charge of the carrier of the current, althoughthis did not indicate whether the current was truly fluid or particles. Even beforethe detection of charged fundamental particles, the electron, the theory had beenfully developed for their motion. He showed that in a magnetic field a charge willexperience an acceleration that deflects its motion in a direction perpendicular toboth its velocity, v, and the direction of the field, B, writing this as a force,

F = q

[E + 1

cv × B

]

including an electric field, E , for a charge q. For a deviation from parallel motion,this force produces an orbit around a region of constant magnetic flux, Faraday’sfield line. This is true whether the field is steady or depends on time and/orposition. The coupling will be the charge, q, and the particle will therefore executea harmonic motion around some direction defined by the field while it moves as wellfreely along the field if initially so directed. Lorentz then showed that the magneticand electric fields can be transformed one into the other by a change in referenceframe in a way that preserved the wave equation. At the end of the nineteenthcentury, Lienard and Weichert introduced the time delay when computing thepotentials of moving charges, completing the electrodynamic program begun byHelmholtz in 1881. The principle is that we don’t see a charge where it is now butwhere it was some time in the past that depends on its distance and the speed of thesignal, the speed of light. Thus there is always a time delay between the cause—themotion of the charge—and its effect—the acceleration of a neighboring charge.This delay produces the emission of an electromagnetic wave and changes respondwith a phase shift. This means the motion of the source must be included and leadsdirectly to our next topic, the theory of relativity.

NOTES

1. A complication was that the effect of the Earth’s magnetism was always present andOersted knew no way to shield his detecting needle from its influence but the measurementswere sensitive enough to at least show the effect.

2. Julius Robert Mayer had already written a more speculative, although also fun-damental, work on the connection between force and work in 1842 but it lacked themathematical precision and scope of Helmholtz’s paper. At the time he composed it,Helmholtz was unaware of Mayer’s work. When he later became acquainted with it, in1852, he acknowledged Mayer’s priority in the basic idea of energy conservation. Joule,Thomson, and Tait, among others, disputed this priority for some time, in a manner typicalof the nineteenth century, but Tyndall later became the “English champion” of the workand Mayer was awarded the Copley medal of the Royal Society near the end of his life, inthe 1870s.


3. The task of setting up standards for electromagnetic measurements and devicesoccupied much of the late nineteenth century, extending well into the twentieth century(finally, after 1940, a consistent set of international standards were defined but even thesechanged their values with the shift from cgs-Gaussian to MKS-SI units in the 1960s.

4. In fact, Maxwell had taken years to edit and oversee the publication of Cavendish’sscientific papers.

5. As an aside, you should bear in mind the whole of electromagnetic theory developedin Maxwell’s hands as a continuous medium. When he speaks of charge he means a fluid, orfluids, that are intermixing and displacing. There is no elementary unit of massive particle,indeed although a confirmed atomist Maxwell was not able to formulate a particle theoryof electricity nor did he require one. It’s very tempting now to reinterpret this in terms ofcharges, you know these too well from a century of pedagogical production in the moderncurriculum was a treatment . The introduction of a displacement current now meant thatthe magnetic field originates not only from the free movement of the fluid, which can afterall be in steady state, but also the specific time dependent reaction of the strained ether.This happened within a medium, the ether for “free space” or a magnetizable substance,the qualitative description of which is that “a change in the strength of the magnetic field,in time, is determined by the displacement current plus the free current.”

6. Testing this prediction, which actually is best understood in an astrophysical contextproduced an amusing sidelight during the late nineteenth century. For some time, a debateraged about whether the effect had been directly observed using a rotating vane the sides ofwhose blades were painted alternately black and white. The idea was that reflection shouldproduce a greater momentum transfer than absorption because of the impulse felt whenthe wave direction is reversed coherently. Thus the vanes should rotate in the sense of thelight colored surfaces. The opposite was seen, however, and this was finally understood asa thermal effect in the not quite perfect vacuum of the evacuated bulb. But it illustrateshow a fundamental idea transforms into a laboratory test that may not be well specified atthe start.


8

THE RELATIVITY OF MOTION

In the last few days I have completed one of the finest papers of my life. When youare older, I will tell you about it.

—Albert Einstein to his son Hans Albert, November 4, 1915

Electromagnetic theory achieved a unification of two forces but at a price. It requi-red a medium, the ether, to support the waves that transmit the force. But this takestime and there is a delay in arrival of a signal from a source when something ischanging or moving during this time interval. How do you know something is mov-ing? Remember that Aristotle used the notion of place within space or memoryto recognize first change and then motion. So there the motion is known by compari-son to a previous state. But this isn’t the same question that recurs in the seven-teenth century. According to Descartes, we have to look instead at the dispositionof bodies relative to which motion is known. These aren’t the same statement,although that would again divert us into metaphysics. Instead, we can look atwhat this means for the force concept. As Newton was at pains to explain, inertiarequires two quantities, velocity and mass, to specify the quantity, momentumthat is conserved. While mass is both a scalar quantity and a primitive in thedynamical principles, the momentum is not. It requires a direction and therefore,if we identify the inertial state as one moving at constant velocity, we need toknow relative to what we take this motion. This is where we begin our discussionof the revolutionary developments at the start of the twentieth century: time, notonly space and motion, is relative to the observer and dependent on the choice ofreference frame and, consequently, has the same status as a coordinate becauseof its dependence on the motion of the observers. It follows that the notion offorce as defined to this point in our discussions must be completely re-examined.Since force is defined through acceleration, which depends on both space andtime, without an absolute frame the whole concept of force becomes murky. This


is the problem with which Albert Einstein and Hermann Minkowski started andwhere we will as well.

THE RESTRICTED, OR SPECIAL, THEORY

Galileo, in the Dialogs, used a principle of relative motion to refute the argumentsof the contra-Copernicans regarding the orbital motion of the Earth, a notion thatis now referred to as Galilean invariance. If we’re in a moving system, and for themoment we’ll assume it is inertial in the sense that it moves without accelerating,any measurement we make of forces in that system will be unaffected by its motion.Otherwise stated, there is no way we can find the motion of the system by forcemeasurements on bodies in the system. The example he used, dropping a body onthe Earth, was based on simple low velocity experiences that are perfectly validas a first approximation. Thus the location of a body moving in an “enclosure” isseen by a stationary observer as a sum of the two motions. Calling x the positionin the stationary frame and x′ that in the frame moving with a constant relativespeed v, the relation between the two descriptions of the motion in a time intervalt is x′ = x − vt . The time, t , is the same for the two observers, a universal rateof ticking of a cosmic clock that they both agree to use. This could also be twoidentical pendula; whether on a moving Earth or an observer at rest with respectto the fixed stars, the pendulum will swing identically and therefore the onlyway to tell which frame the measurement was in is to ask some other questionthan what is the law of the pendulum. You know from our discussion of theCoriolis effect that there is a way to distinguish these systems, the one that is notmoving inertially will see the pendulum precess with a period related to the rate oftranslation of the moving frame relative to the inertial observer and therefore willexperience a force. But the time interval is the same for the two observers. Theforce law is altered by the addition of these inertial forces (to use D’Alembert’sconstruction). This was already included in Newton’s laws of motion since theequivalence of states of rest or inertial motion guarantees Galilean invariance.Einstein’s conception was both more radical and simpler. More radical becausehe allowed the time to be also something varying for the two observers, and simplerbecause he asserted in an even stronger form the basis of Galilean invariance: thatthe physical laws are identically measured by any two inertially moving observers.This second assertion doesn’t obviously yield the first as a consequence exceptfor one additional assertion, concordant with experiment. The two observers mustagree, somehow, on their relative motion. That they are moving can be assertedrelative to another, third, body but that gets us into an endless regression. Instead,we use as a fact that the communication takes place by signals at the speed oflight and this (finite) speed is independent of the motion of the observer.

It began with a question from Einstein’s youth. At age 16, he pictured lookingat a clock while moving away from it. Since you see the face and hands becauseit is reflecting, or emitting, light, if you move slower than the light you will seethe hands move (now we would say the numbers change on a digital clock). Butif you were to move at the speed of light, if you looked at a watch on your armyou would see the hands moving but you wouldn’t when looking at the stationaryclock. It would appear to you that time was frozen yet in your moving referenceframe you would be aware of the passing of the hours. Now, with the result that the

The Relativity of Motion 161

speed of light is actually independent of the observer, Einstein imagined how toextend this with a simple construction. Imagine, instead, that the moving observeris holding a mirror. The image she sees is a reflection of a light beam. His questionbecomes “does she see her face if she’s moving at the speed of light?” If thespeed of that light is independent of the motion of the observer plus mirror, theanswer must be “yes.” But he now asked what someone sitting on the groundwatching her whiz by will see. Since the speed of light is the same, there must be adifference in the time it takes for the signal to traverse the two apparently differentpaths. Let the propagation be perpendicular to the direction of the moving frame’svelocity and call that v. The moving observer measures a time interval �t to goa distance L. The stationary observer sees the same speed but following a longerpath,

√[(2L)2 + (v�t )2)]. This must mean they don’t see the same time interval

and the difference depends on the relative speed of the moving system. This isthe concept of time dilation, that the time �t ′ = �t/

√(1 − v2/c2). Further, if

we ask what the two observers will measure for the length of the path, it must bedifferent because the speed, which is a ratio of length to time interval, remainsthe same. The effect is completely symmetric—as long as the relative velocity isconstant neither observer can tell which of them is moving, only the relative speedmatters. This is the origin of the concept of relativity, the modifier “special” or“restricted” in the usual name for the theory refers to the requirement of inertialmotion. Time and space cannot be absolute, the most important change in themechanical picture of the universe since the introduction of inertial motion.1

The Maxwell equations were the framework in which Einstein first discussedthe special principle of relativity. When moving through a magnetic field, a chargeexperiences a force as we’ve discussed, that depends on its velocity. In the co-moving system, however, this force doesn’t exist. If you move with the current theelectric field seems to be static. So it would seem the equations of motion changetheir form depending on the reference frame. If so, do the equations describingelectromagnetic phenomena also change? There is nothing special about two sys-tems, it seems, in one of which the charge moves and the observer remains at restor the other in which the observer moves past a stationary charge. If the two framesare formally, mechanically, equivalent for inertial observers, you would expect thatthe Maxwell equations for the electromagnetic field should also be transformableto a general system that is independent of the specific reference frame. Thus thetitle of the first paper on the theory, On the Electrodynamics of Moving Bodies, inwhich he sought to extend the investigations of 1892 by Lorentz on the same topic.The fundamental difference was epistemological. While Lorentz had assumed anether, Einstein approached the problem as a transformation between observers. Itis in this difference in perspective that we see the profound nature of the programthat would eventually lead to a new theory of space and time and forces.

There was also an empirical basis for Einstein’s assertion of the constancy ofthe speed of light. The paradoxical observation by Albert A. Michelson, in 1881and his repetition and refinement of the experiment with E. Morley in 1887, wasthat there was no trace in the variation of the speed of light for a device in relativemotion with respect to the ether. The apparatus was an interferometer consistingof two perpendicular arms that split a beam of light and then, on recombination,produced interference fringes. The phase of the light at recombination dependson the time of travel across the two arms and, so the reasoning went, because of


the relative motion there would be a difference in the time of flight along, andperpendicular to, the direction of motion of the beam. This was not detected to avery high accuracy. It was already known, however, that there is a displacementdue to the finite speed of light, the aberration of starlight discovered by JamesBradley in 1725 and explained as the relative motion of the Earth with respect to adistance light source. If you’re watching a particle, this isn’t a problem because it’sjust the vector sum of the components of the velocity. An electromagnetic wave,however, moved in Maxwell’s theory through a medium and it is very strangethat there was no effect at all. This was explained dynamically by Lorentz, whosought to maintain a Galilean framework in which time is absolute and only spacemotions are relative, and postulated a physical change in the length of the arm inthe direction of the motion. This may seem an ad hoc approach but for Lorentz itwas a necessary consequence of the electromagnetic strains produced by the etherwhen a body is in motion. Einstein had, instead, substituted a new central axiom,within the kinematic theory, of the constancy of the speed of light measured byany observers in inertial systems in relative motion. Since the speed of light, c,is a dimensioned quantity, to assert its independence of reference frame impliesa fundamental connection between the way two moving observers agree on theirdefinitions of space and time. We can take this as an empirical fact. The secondaxiom, causality, is a bit more subtle. For the entire history of physics up to thebeginning of the twentieth century this merely implied a semantic relation betweencause and effect. But for Einstein, this was extended to a formal mathematicalstatement, although this only became clear with Hermann Minkowski’s (1907)geometrical representations, that for what is actually a unified spacetime, thecausal spacetime interval measured by two observers between events a spatialdistance dx apart and separated by an interval of time dt is d s2 = c2dt2 − dx2,is constant independent of the motion of the observer. This is a generalization ofthe geometric concept of distance: the motion is in time as well as space and thetotal distance covered is in the spacetime. There is a very important difference,however, with the usual Pythagorean theorem, the sign of the constituent terms.Instead of being a simple sum of squares, the space and time components sumwith opposite signs, so if you are moving with the speed of light the space intervalis essentially the same as the time (with c being a dimensioned proportionalityconstant). The next step was Einstein’s extension of Galilean invariance, for whicha position in a stationary reference frame can be related to that in one that is movingat constant velocity by x′ = x − vt , if the relative velocity v is sufficiently slow.The direction of the motion selects which coordinate requires transformation. Sincethe time is relative, by extension so is the space interval. Einstein then postulateda unique, linear transformation law that would combine space, which I’ll denoteas the coordinate x and time t and then asserting that the interval measured in thetwo frames is the same, (ct )2 − x2 = (ct ′)2 − (x′)2, he obtained the same resultthat Lorentz and Fitzgerald had found from a completely different analysis of theMichelson-Morley results:

t ′ = γ (t − vx/c2)

x′ = γ (x − vt ),


where γ = 1/√

(1 − v2/c2), now called the “Lorentz factor.” There are no actualchanges in the structure of the body because of the motion but only apparentchanges due to the causality condition and the constancy of c. The two observerssee things differently though neither is “correct” in an absolute sense. Einsteinadded the concept of simultaneity, the requirement that two observers can agree ifthey are relatively appropriately situated. By asserting its possibility, he introducedan objective reality into a relativistic construction, the event does happen and it’sup to the two witnesses to agree on how to coordinate their observations. A strikingconsequence is the change in the addition law for relative motion. Instead of theGalilean form, he found that if a speed v is seen in a frame moving with a velocity u,instead of the relative motion being v + V it becomes v′ = (v + u)/(1 + uv/c2).In all cases, this is a direct consequence of keeping the constant speed of light forall observers—no signal can propagate causally faster than c and two observerswill always, regardless of their relative frame motion V , see the same value forthe speed of light. It then follows that no body with a finite mass can travel at aspeed c.

Let me digress for just a moment. The fundamental picture of the universecontaining an absolute space and time relative to which, independent of theobserver, all motion can be measured had not changed for two thousand years. Ithad been elaborated differently through the centuries but had even underpinnedNewton’s development of mechanics. The idea that all physical experience is onlyrelative and that there is no longer a separation between the motion of an observerand the observation was a rupture in the foundations of physics. Every change inideas about motion has provoked a change in how force is understood but Einsteinhad now replaced space and time with a new entity, spacetime, so designatedby Minkowki. To pass from kinematics to dynamics, however, required anotherpostulate, again related to the speed of light. The two observers must be able toagree on the measurements they’re making. That requires some way to establishwhat it means to do anything simultaneously. At some moment in space and timethe two observers must be able to coordinate their measurement device together.Further, their measurements should be independent of which frame they’re in. Thisis the principle of covariance, that the physical laws are independent of the state ofrelative motion of the observers. But this comes with a price. If in one frame a forceis applied to an object, in the other it will not seem to be the same. Accelerationresulting from the action of a force depends on the mass of the affected object. Butacceleration is the change in a velocity in an interval of time and a force is thechange in momentum in the same interval. If the length and time are changing tomaintain constant c, then one observer sees a different acceleration than the otherand must conclude that the momentum is changing differently. Even more clearly,if a body is accelerated by a known force, then as its speed increases so does thediscrepancy between the force applied and the observed acceleration. This impliesthat the mass is increasing since the acceleration decreases. The only thing wecan define is the mass at rest, once moving the inertia of the body increases. Thelimiting speed is c and therefore when that is reached the body would not accelerateat all, it’s inertia would be effectively infinite. This is what Einstein presented inan amazingly short companion paper to his electrodynamics, “Does the Inertia of aBody Depend on its Energy Content?.” He showed that the new dynamics required


a redefinition of mass. What we measure is m = m0/√

(1 − v2/c2), where m0 isthe rest mass. But defining “rest,” remember, is relative so this holds equally to thetwo frames when the coordinate transformations are applied. He then showed that atslow speeds, the kinetic energy we measure is 1

2 mv2 but with an additional term,a zero point called the “rest energy” that had been previously unsuspected, and isnow the iconographic statement of the equivalence of mass (inertia) and energy,E = m0c2. Because of the relativity of space and time, energy must also be relativeand what you think is the kinetic energy is actually different than what you canextract from actual collisions. If this applies to moving systems, Einstein realized,it also applies to any accelerated motion, in other words even if something isbound in a potential you measure less mass in a static case than you will be ableto extract dynamically.

GRAVITATION AND GENERAL RELATIVITY

Inertial motion is very restrictive and Einstein understood that the special theorywas only the first step. The struggle to complete this program took over 20 years,culminating with the publication of the field equations for gravitation, GeneralRelativity Theory (GRT), in 1916. The generalization of the relativity principle toaccelerated motion provided the first fundamentally new conception of gravitationbased only on motion and geometry. At the end of the nineteenth century, BaronEotvos had demonstrated the equivalence of gravitational and inertial mass. Thisis the same thing as saying that regardless of the force applied the resistanceto acceleration, its mass, is the same as measured by its weight, or that thecomposition of a body really doesn’t matter when you are weighing it. Finally whathad asserted in Newtonian gravitation was experimentally verified. It meant, ineffect, that the program stretching back to Archimedes, the geometricization offorce, might yield an explanation for gravity.

The Equivalence Principle

The equivalence principle was the epistemological core of Einstein’s thinking,the insight that changed how force is conceived. The example he gave, imaginingyourself in a freely falling elevator (sealed and with no information from outside)was a thought experiment in frames of reference. The trajectory of the elevator is acurve in spacetime because it accelerates. No measurement, however, within thislocal system can distinguish the absence of gravity from freefall. Within the framethe motions are invariant to changes at constant velocity and there are, obviously,ways of producing forces as well (think of stretching a spring while inside). Ifyou suspend a weight from a spring and accelerate the elevator upward (insteadof intentionally imagining your impending death), the spring extends when theelevator starts to move exactly as if the weight, or the acceleration of gravity,had instantaneously changed. Schematically, this is the first step to the covariantformulation of any theory of gravity: for Einstein, at least at first, gravitation wasthe result of not being able to freely fall and therefore any physical measurementsmade in one frame should be transformable into any other with just a coordinatechange. In other words, with the appropriate description of the trajectory, you can


be put on a path through a spacetime in which you don’t know about a gravitationalfield. On the other hand, you are actually in freefall, and the gradients in the fieldproduce your trajectory. This isn’t something happening in a source for spacetime,there’s a mass there—somewhere—that is distributed in such a way that youexperience changes in its potential. Think again of the equipotential surfaces.Their normal gradient gives the acceleration but the divergence of this throughspace is the result of a source. No source, no divergence. Unless, that is, there is aunique point source and we’re far enough away from it, or in the proper geometry,that the local effect can be made to vanish merely by displacing our starting pointfar enough away that the divergence becomes infinitesimally small.

Much of this we already know from the approach that started with Green. ThePoisson equation must transform into geometric terms when we enforce this con-dition of covariance, as must the Laplace equation. If we identify the potentialsof the field with the metric on the surface, along with Einstein, then we can saythat the curvature (which is measured by the divergence of the normal gradients)is produced by the mass distribution. We can talk—again—in purely geometricterms about gravitation and completely ignore the individual motions thus pro-duced. This may seem very counter-intuitive since you have, up to this point, beenthinking in terms of Newtonian gravitation. But one of the main features of generalrelativity is to render all phenomena as contrast between inertial and acceleratedmotions and show how to transform between the two.

To extend this idea, imagine now having a ping-pong table placed inside thecab. Again you watch but this time there are two robots that hit a ball repeatedlyacross the table. You check their activities to be sure the ball is perfectly elasticand they’re always hitting with the same stroke. Then at some moment, you seethe ball passing between them not in an arc but directly across the table. By thereasoning of the equivalence principle, we know what has happened: the elevatoris in freefall. In the local frame there is no net force and the motion appears purelyinertial. So even if we have a non-inertial motion, we can transfer into a coordinatesystem that permits solution of the dynamics as if there were no forces, and thenfind how the frame is accelerating and reconstruct the trajectories in our referenceframe.

Gravitation as Geometry

First let’s begin with a simple example. You want to make a map using a group ofsurveyors each of whom is dispatched to a different city. To be specific, take NewYork, Baltimore, Chicago, and Boston. On arriving at their stations, each sets upa local coordinate system, establishing the vertical as the direction perpendicularto the horizontal (which may seem obvious but wait a moment), and measures theangles between her city and two others, for instance the angle measured at NYbetween Baltimre and Chicago. After they have closed this net of cities, theseobservers compare the sums of the angles. For a tri-city set, a triangle, they knowfrom Euclid that the sum should be 180 degrees. But when they do the sum, itisn’t. What are they to conclude about the surface? This problem, posed preciselythis way, occupied Gauss at the start of the nineteenth century and produced a firstpicture of surface curvature: in the measurement is made purely geometrically,


Figure 8.1: Albert Einstein. Courtesy Library of Congress,Prints and Photographs Division, LC-USZ62-60242.

and the result for all possible trianglesdiffers from the expected sum by thesame amount, the surface is curved andthat curvature is uniform. The radius ofcurvature, the degree to which the mea-surements depart from the Euclidean flatplane, is found by the dependence ofthe difference in the sum on the met-ric separation of the points of referencein the global system. Any rotation of thetriangles, any shuffling of the measure-ments, will produce the same answer ifthe surface always has the same topol-ogy, the same “form” of curvature. As afurther point, if two observers establish abaseline and then translate perpendicu-lar to that, always carefully establishinga baseline after some parallel displace-ment without checking the distance, theywill either relatively converge or diverse.If they ultimately converge, the surface ispositively curved and the simplest exam-ple we know is a sphere. If they diverge,the curvature is negative and, geometri-cally, we can imagine it is saddle-shape.

Now we ask a different question, how does each surveyor establish the vertical.They might simply put a piece of paper on the ground and find the line that makesan angle of 90 degrees to it and doesn’t lie in the plane. But more likely, theywill use a plumbline. Now they might notice that these two don’t match and we’vereturned to the concept of the potential as a surface. The normal component of thepotential gradient, which defines the local gravitational acceleration, depends onthe local distribution of masses. This was the result of the Maskelyne measurement,

trajectorytrajectorymass

moving massmovingmass

Newtonian gravitationaldescription

General relativitydescription

locationof mass

equipotentials

Figure 8.2: A cartoon comparison of the classical (Newtonian) and relativistic descrip-tions of the deflection of a body by another mass. In the left panel, the trajectory ischanged because of a force. In the right, one can describe the motion along a curvedspacetime (here shown schematically as a potential well but the motion is really infour dimensions, the time changes along the trajectory).


the deviation of a plumbline because of the vicinity of perturbing masses (i.e.,mountains in his measurement). But if the surface is apparently level, in thesense that there are no local variations in altitude, we would conclude that thevertical deviates because the Earth’s substructure is not of the same symmetry aswe assumed for the curvature based on the geometry. So we have a way to seehow curvature defined by a measurement of the potential gradient relates to thatdefined by the local tangent plane, the surface geometry.

Let’s go one step farther. If we again take two observers, one of whom staysfixed while the other walks in a straight line, that is constantly noting that thestationary observer always stays at the same bearing relative to an instantaneouslyestablished coordinate frame (again, measuring the local vertical and the anglerelative to the original bearing of the trip), the two may notice that the two verticaldirections are not translating parallel to each other. That is, the normal vectormay rotate in the plane perpendicular to the surface. If the two stay aligned, thesurface is obviously fat in the euclidean sense and we are simply moving in auniform straight line. But if the angle changes, increasing in a positive or negativesense, the surface is curved. Again, we can measure this using a plumbline butthat introduces a new feature, the local gravitational field may not be followingthe surface as we define it by the path. The parallel translating observer thinksshe is walking a straight line. In other words, if she maintains constant speed anddirection he is moving inertially. But the stationary observer sees that this isn’t so,he concludes there has been an acceleration even if the movement is at constantspeed (for instance, the two can exchange a signal at fixed frequency to checkthat there has been no change, hence no acceleration measured by the Dopplereffect) since the two normals are rotating. Further, if the moving observer thentranslates around a closed circuit always maintaining the same heading, withoutrotating the coordinate system, then at the end of the circuit when the two arereunited there will be a rotation of the coordinate patch with which the movingobserver recorded the original directions when starting the trek. In moving arounda closed curve—that is, an orbit—around a central mass in the two body problem,the orbiting body is moving at constant speed and parallel translating. But this isclearly an orbit because if we translate the velocity vector back to the starting pointit will have rotated relative to the radial direction by an amount that depends onthe angle through which it’s been displaced around the orbit. The change, Newtonrealized, will always be in the radial direction and, since by the second law anyacceleration is in the same direction—and with a magnitude that depends on—theforce producing it, we interpret the change in the velocity vector as a result of aforce.

This is the equivalence principle in another guise. Any acceleration is inter-preted—based on this geometric observation—as a force and we have now com-pleted the analogy that Einstein and Marcel Grossmann (1878–1936) achieved intheir 1913 paper on gravitation, Entwurf einer verallgemeinerten Relativitatstheorieund einer Theorie der Gravitation (“Outline of a General Theory of Relativity andof a Theory of Gravitation”). Because this is a local effect, constructed by takingpieces of the path and moving them, they had constructed a differential geometricdescription of the local curvature that is equivalent to a force. The deviation ofthe normal gradient to the equipotential surface is the same as a differential force,


this provided a description of gravitation: a gravitational field can be representedusing differential geometry as a curvature. Since this happens in both space andtime because of the relativity of a spacetime, this is a curvature in four dimensions,the basis of the General Theory of Relativity for gravitational fields.

The direct influence through the Einstein–Grossmann collaboration wasBernard Riemann’s 1857 thesis On the Hypothesis at the Foundation of Geom-etry, a topic chosen by Gauss at Gottingen. In this work, Riemann introducedthe idea that a differential expression for distance can be generalized for a non-Euclidean surface or space, maintaining the same form but allowing the metricaldistance to depend on position. In Grassmann’s or Hamilton’s language, this is thesame as saying the scalar product—that is, the projection of two vectors onto eachother—depends on the position in the space. The projector is dimensionless if thetwo vectors are unit distances and, instead of the Pythagorean theorem, that squareof the total of small displacements is the sum of the squares of the componentdisplacements (hence using dx for each component),

d s2 = dx · dx.

Riemann generalized this using a projection factor, the metric, which is a matrixthat also includes the displacements in the plans and not only along the individualaxes,

d s2 =∑i, j

gi j d xi dx j ,

where we sum over all possible values of the two indices i and j (for instanceg11(dx1)2 + g12dx1dx2 + · · ·). In words, this says that for two infinitesimal dis-placements, instead of being the sum of the squares, the distance may changedepending on how the two distances displace parallel to each other. The change inthe properties of the surface can then be determined by examining how the metric,this projector, changes as we move around the space. The relation between thismetric and the surface (or spatial) curvature was made more precise by GregorioRicci-Curbastro (1853–1925) and his student, Tullio Levi-Civitta (1873–1941),during the first decade of the twentieth century. Specifically, Levi-Civitta wasthoroughly familiar with dynamics, especially rigid body motions, and introducedthe idea of parallel displacement to describe motion on generalized curves. LuigiBianchi (1856–1928) further developed this “absolute calculus” by finding therelation between the scalar quantity—the curvature of the surface, and the localmeasurement of the deviation of the vectors by this transport.

Grossmann, who had been a fellow student with Einstein at Zurich and was now aprofessor of mathematics at the university there, knew this part of the mathematicsand Einstein had the intuition for how to apply it. Their principal joint paper, theEntwurf paper of 1913, actually consisted of two separately authored parts, one onthe methodology (Grossmann), the other on the physical setting (Einstein). Whilestill incomplete, this was the crucial start to what proved to be a difficult, fouryear endeavor for Einstein alone. Let’s return to our surveyors. Recall we used the


Doppler shift to establish that the speed is constant between the two observers. Itdoesn’t matter, really, if one is truly stationary or not, they are simply in relativemotion as in the Special (restricted) Theory of Relativity. Now in sending thesignal, if the rotation of the normal vector is due to a gravitational field, anyacceleration must produce a change in the length and time intervals. If the speedof light remains constant, the alteration of the normal gradient to the equipotentialswill produce a shift in the frequency of the signal, if it emerges from a strongergradient, if the normal direction changes more rapidly, the signal will be receivedwith a redshift—a downshifting of the frequency. It also means, because the pathof the light is constrained to maintain constant speed, there must be a curvatureto the path even in a vacuum irrespective of the nature of the light and even if thepulse has no mass. In Newton’s conception of light, a particle with a finite mass, agravitational field produces a deflection when it isn’t along an equipotential. Butit’s a different effect. If light is photons, and these are massless, their constantspeed requires again a deviation but with a value a factor of two different fromthe Newtonian prediction. Einstein realized this even before the theory was in itsfinal form, as for the redshift, and suggested after discussions with an astronomerat Gottingen, H. Freundlich, that this might be observable as a deviation of thepositions of stars near the limb of the Sun during a total solar eclipse. Now we canask what is the source for this change in the trajectory? In the two body problemwe identify it with the mass of the two orbiting bodies. The closer they are togetherthe greater the gravitational acceleration, therefore from our new perspective thegreater the curvature of the spacetime. Since we already know from the restrictedtheory that the inertia and energy are identical, the source can be generalized tothe total energy density.

You can now see why Einstein followed this path. In a gravitational field allbodies experience the same acceleration regardless of their mass. The differenceis the force. They follow identical trajectories, that is freefall, and we can alwaysfind a frame of reference—the moving observer—for whom the principal gravi-tational field of a central body vanishes. But there is also an effect of the tidalcomponent, the higher order derivatives of the gravitational field, that never vanishregardless how small we make the region. Thus for a strictly local measurementit is reasonable—the elevator gedanken experiement—to imagine an observer infreefall feeling locally no gravitational acceleration in whose reference systemthere is only inertial motion (or effects produced strictly by interactions withinthat frame). This is motion along the normal gradient to the equipotentials. Alonga path we can define an ordered sequence, a single monotonic parameter thatdescribes the steps. Since the observer receives and sends signals to others, thereception times depend on the spacetime as a whole but she can look at her watchand note the times. This is the proper or comoving frame, the time in this frameis therefore an adequate affine parameter, the sequence. The observer defines anymotion as covariant relative to the contravariant background.

We return to Green’s conception of the field. A fluid mass adopts a shapethat corresponds, absent internal circulations, to the equipotential surface. Thismay be substantially distorted from a sphere depending on whether the mass isrotating or isolated. In a close binary system, for instance, the Roche solutionto the potential produces a cusp in the shape when the local gravitational forces


balance. This can be easily seen in the translation of a mass over the surface.On any equipotential the motion will remain constant so this is inertial, butin a non-inertial reference system such as a rotating body, the normal cannotremain constant on the equipotentials. In a more complex mass distribution, thelocal curvature thus reflects the distribution of the masses and we have finallyfinished. Instead, in general relativity, the geometry is the field. This is whyEinstein persistently referred to the metric as the field itself, with the curvaturetaking the role of the field equations. The field equation must reduce to theNewtonian limit, therefore to the Poisson equation, when the fields are sufficientlyweak.

Orbital Precession and the Two Body Problem

Planets orbiting the Sun follow closed trajectories, at least in the original formof what we’ve called the Kepler, or two body, problem. The period dependsonly on the angular momentum of the body and its distance from the centralmass. Now, remaining within the frame of special relativity, we think about thevelocity of the planet. When it is near the Sun, for example at perihelion, it’smoving faster than at aphelion. Comparing two instants, the inertial masses willnot be identical and the angular momentum can’t be the same either. Althoughit’s stretching the limits of special relativity to describe it this way, to a trulyinertial, stationary observer watching this the planet will appear to move as ifaffected by some other force, or by something other than in inverse square centralattraction. The orbit, in fact, cannot be strictly periodic and must precess, exactlyas though some other body were perturbing its orbit. The change, due to therelativistic effect, depends on only one quantity, the orbital speed. Since this isalways small compared to c, we can be sure the effect will not be very large. Butneither does it absolutely vanish and thus, after all the planetary perturbationsin a real solar system are accounted for, there should be a residual apparentforce that somehow is related only to the central body. Throughout the nineteenthcentury, progressively more precise measurements of the motion of the planetswere achieved. Because the era of telescopic measurement had, by the end of thecentury, spanned more than 150 years, for the shortest period planet—Mercury—it was clear that a cumulative apsidal motion that could not be simply explainedby either measurement errors or the perturbations of the planets was present as aresidual in the data. Small though it was, about 43 arcseconds per century, it wasnonetheless far in excess of the typical measurement errors of a few arcseconds,and it was increasing secularly. It was natural to try the same explanation thathad proven so spectacularly successful in the discovery of Neptune, that anotherbody was present within the orbit of Mercury whose gravitational perturbationwas strong enough to affect Mercury without altering the orbits of the neighboringplanets. Not surprisingly, it was Leverrier who attempted this and predicted theorbit based on the Mercury anomaly of a body, subsequently named “Vulcan,” oflow mass and very short period. That the body was sighted several times duringthe last decades of the century is not important, it doesn’t exist and that aloneeventually ruled out this obvious explanation. One other possibility remained,


the same one invoked by Newton to explain the lunar apsidal motion: the Sunmight not be spherical. As we have seen, the gravitational potential of a sphereis different in its radial dependence than a spheroid. Regardless of the origin ofthis presumed eccentricity of the Sun, were it homogeneous and differed from asphere by only a few arcseconds (about 10−5 in relative radial distortion) thiswould produce an alteration of the gravitational field at Mercury sufficiently largeto account for the excess. Again, the observations firmly ruled out any deviationof the proposed magnitude. Lacking these explanations, Simon Newcomb—thendirector of the U.S. Naval Observatory and the leading celestial mechanician ofend of the nineteenth century—proposed a drastic solution, that at short distancesthe force law showed a small departure from 1/r form of the potential, of order onepart in ten thousand in the exponent. This has the same effect as a nonsphericalSun but a totally different origin that, he admitted, was purely a hypothesis. Byitself, the suggestion shows that the concept of gravitation was still not quiteconsolidated since all other successes in celestial mechanics should have servedas sufficiently strong tests of the force law to rule out this speculation a priori.

The explanation is, instead, a product of the relativistic generalization of gravi-tation. To state this in the language of GRT, in the vicinity of a massive body—forinstance the Sun—the curvature of spacetime is changed from that of a classicalGM/r potential field to one only slightly different. Since the space is no longerflat, an orbit doesn’t close even around a point mass and, as a result, an ellip-tical orbit slowly precesses. Recall the orbiting particle is in freefall not onlymotion though space in time, but motion in spacetime, and therefore both thetime and space intervals it experiences are different from those of an observerat rest on the central mass. Einstein realized this orbital precession as the firstconsequence of the weak field limit in the final version of GRT using the calcu-lation from a completely different context. In 1920, Arnold Sommerfeld appliedthe relativistic correction to the Bohr atomic model in which an electron orbitsthe nucleus at nearly the speed of light. For such a particle, although its restmass is constant its momentum is not and therefore neither is its orbital angularmomentum. Its inertia increases when “perinuclear” (near the force center) anddecreases when “apo-nuclear.” The variation produces a precession even for thetwo body Kepler problem.2 The change in the field is enough to precisely producethe already known anomaly, a “retrodiction” that proved decisive in Einstein’slabors.

The Schwarzschild Solution and Singularities

The field equations were presented in their final form in Einstein’s paper of 1916.Their solution for a point mass was obtained by Karl Schwarschild (1873–1916)in the same year, almost immediately after their publication. It was a tour-de-force calculation made possible by a reasonable simplification: for a sphericallysymmetric point mass the problem is one dimensional. The time (g00) and space(g11) components of the metric depend on the radial position, which is obviouslytrue using the picture of equipotential surfaces precisely because we have aspherically symmetric system. The same is true for the curvature. Because the


mass is pointlike its external field is a vacuum so the classical limit for thefield equations is the Laplace equation. Schwarzschild then was able to solve thelinearized problem and found:

g00 =(

1 − 2GM

c2r

),

g11 = 1/ (

1 − 2GM

c2r

).

The angular coordinates define motion on a sphere of constant radius. This isbecause the only change that matters is crossing from one equipotential surface toanother, not motion confined to such surfaces; circular orbits do not spontaneouslybecome elliptical. For Mercury, at a distance of about 1/3 the distance of theEarth from the Sun (about 100 solar radii), this amounts to a deviation in both thetemporal and radial components of only about 5 × 10−6 from a perfectly flat metric.With an orbital period of about one quarter year, this amounts to a change of aboutthe right order of magnitude for the nonclosure of the orbit per cycle. Einstein hadalready obtained this result before arriving at the final form of the field equations sothis agreement was important. There are two additional consequences. The changein the time component implies a redshift when light emerges from the body andheads toward the great beyond, the redshift (infalling light will, instead, experiencea blueshift but that is not visible by a distant observer). A third consequence is thebending of light trajectories at any distance from the central mass: since the Sunhas a finite size, its limb defines the point of closest approach for an “orbit of light.”

Schwarzschild’s solution, however, introduced a puzzle. Green had initially in-sisted that the distribution of charge on a conducting surface remain non-singular,that it be everywhere smooth (and therefore continuous so the derivatives also existeverywhere and there are no infinite forces). Now we have a strange consequenceof the relativistic equations: at a distance R� = 2GM/c2 from a point mass, thespatial metric component becomes infinite and the time component vanishes. Inother words, viewed from afar, a freely infalling observer will simply appear to hangat some distance from the central point mass and disappear in as a progressivelymore redshifted image. At first thought to be merely an effect of a poor choiceof coordinate representation, it is now understood that the Schwarzschild metricpossesses an event horizon from inside of which it is impossible to receive a signalalthough the infalling particle experiences nothing unusual (except for the tidalacceleration resulting from the locally enormous curvature). The curvature doesn’tvanish even though the metric becomes singular, hence the spacetime can be con-tinued smoothly into the center, although from the publication of Schwarzschild’spaper to the early 1960s this was thought to be a mathematical artifact and therewere several attempts to remove it by different coordinate representations. Thecontinuity of the spacetime was finally demonstrated by Krushkal and Szekeres in1965, long after the original derivation of the metric, and the metric for a rotatingisolated mass was obtained by Roy Kerr in 1963. With these two solutions, theimportance of the singularity and the event horizon were finally recognized and anew name was coined by John Wheeler, a “black hole.”


Gravitational Lensing as the Map of the Potential

The most celebrated confirmation of GRT came in 1919, the detection of thegravitational deflection of starlight at the limb of the Sun observed during a solareclipse. But that is only a part of the story. Newtonian mechanics also makessuch a prediction, the trajectory of a body moving at the speed of light (which isnot a speed limit in classical mechanics) should be about 0.9 arcseconds at thesolar limb (one solar radius for a one solar mass body). Relativity, instead, makestwo fundamental modifications, both qualitative, that predict to a value twice theNewtonian deflection. Since time is relative, the frequency of the light is changedon the two legs of the path, first being blueshifted on ingress into the gravitationalpotential and then redshifted on egress. The net change is, therefore, zero butthis changes the effective inertia of the light. The second is the change in themomentum as a function of position in the spacetime, depending on the energyof the light since p = E/c. This is a direct measurement of the structure of thespacetime.3 This optical consequence of general relativity is more, however, thanjust a test of the theory. It provides a means to image the potential, to visualizethe correspondence between the force and the geometry. Unlike a normal lensingdevice, the vacuum has a constant, uniform “index of refraction” and thereforeany change in the trajectory of light is only because of the change in the spacetimestructure, not its material properties. Recall that Maxwell had assumed the etherhas a constant dielectric constant and magnetic permeability. In the transitionbetween two media, or if either of these physical parameters varies within aregion, the path of light changes because of dispersion. A wave in vacuo doesn’tdisperse but follows a path of least action, a geodesic, and therefore traces thecurvature. Now if instead of a point source we view an extended object througha distribution of masses, the lenser is more complex. Two or more paths mayconverge at the observer. This has been observed in a setting Einstein did notimagine, but Fritz Zwicky did in the 1930s. The distance between galaxies ishuge, of order hundreds of millions to billions of light years, but the sizes ofclusters of galaxies are not that much smaller, about one to ten percent. Thus, adistant object whose light crosses one of these clusters is, in effect, still in the nearfield of the lenser and a focused image can be produced by convergence of raysfrom more than one direction. The resulting image is, to be sure, very distortedbut completely characteristic: since there is no net redshift because although thelight must emerge from the vicinity of the cluster, it also entered from a distance,the spectrum of the light is unchanged while its path deviates. The images shouldhave exactly the same spectrum, a wildly improbable vent if one is really seeingmerely a chance superposition of two distinct cosmic emitters. Such lenses on largescale have been observed, now almost routinely, since the discovery of the firstlensed quasar, 1957+256 (the “name” is the celestial position) in 1979. Lensingby individual stars—microlensing, the closest thing to the original test proposedfor GRT—has been possible since the mid-1990s but this is only because of themotion of stars in front of a more distant background and requires a differentobserving technique. Here the motion of the lenser in front of a densely populatedbackground of stars, for instance toward the bulge in the center of the Milky Way,produces a ring that crosses the Earth’s orbit. The brightness of a background


star changes as it is lensed in a symmetric way over a timescale of several days.The technique has even detected close, unresolved planetary companions of thelenser since any mass, no matter how small, will produce a deflection and leadsto multiple peaks in the brightness variations of the background star.

Gravitational Waves: A New Wrinkle

Classical gravitation has no time dependence. If bodies move the potential changeswith their re-ordered distribution. But in GRT, there is a new feature that is againdue to the finite speed of any signal. A change in the symmetry of a body, or ofthe distribution of mass in an ensemble, alters the structure of the gravitationalfield in time, not only in space. You can think of this as a change in the curvatureof the spacetime, which propagates away from the source and carries energy(and, momentum). The emissivity of the configuration (we’ll say system but thiscould also mean a single body whose shape changes) depends on the rate atwhich the distribution changes. Eddington, in the 1930s, realized that this is anecessary consequence of the new theory of gravity but it is extremely weak (madeaccessible only by the possibility that enormous, of the order of stellar, massesmaybe involved.

The Hertz experiment, primitive as it was and as early in the development ofelectromagnetic theory, succeeded because the coupling between electromagneticwaves and matter is so strong. For gravity, in contrast, the coupling is nearly fortyorders of magnitude smaller and, it would seem, the detection of such waves as a testof the theory is a hopeless task. In the 1950s, however, Joseph Weber suggested away to do this using massive cylindrical bars that would resonate due to mechanicalstrain produced by a passing gravitational wave. Like an EM wave, a gravitationalwave is polarized in the place perpendicular to its propagation direction. Unlikea vacuum electromagnetic field, however, the oscillation is quadrupolar and has adistinctive signature of strains in two orthogonal directions simultaneously with thetwo senses of polarization being at 45 degrees to each other at the frequency . Inaddition, because of the extremely strong frequency dependence of the generationmechanism, the rate of energy emission varies as 6, only the highest frequenciesneed be sought and the bars can be made manageable sizes. Several cryogenicallycooled bars have been operating since the 1960s in laboratories around the world.Since there isn’t a single, fixed frequency expected from cosmic sources the narrowbandwidth of the bar detectors is a serious limitation. A different mode for detectionwas proposed in the 1980s that uses a Michelson interferometer, much like thatused to study the variation of the speed of light due to the motion of the Earth. Alaser beam is split into two and sent through a pair of perpendicular evacuatedtubes. The beams are reflected from mirrors mounted on suspended masses at theends of the tubes and recombined to interfere. Actually, the beam reflects a largenumber of times from both ends of each cavity before recombination to extendthe pathlength, thereby increasing the sensitivity because when a gravitationalwave crosses the instrument the lengths of the two arms change. The differentialchange in the path is detected by looking for changes in the intensity of theinterference pattern using photoelectric cells. The fluctuations in time of the lightsignal contains the frequency information with which to construct a broadband


spectrum, the current limits being a few kilohertz. Several such instruments,actually full scale observatories, are now operational, the principal ones beinga four kilometer long the Laser Interferometric Gravitational Wave Observatory,LIGO, in North America and Virgo, with three kilometer arms, in Italy. Others arelocated in Australia, Japan, and Germany, and there are plans for constructingsuch detectors in space on scales of millions of kilometers. Combined with thebars these form a worldwide network. But unlike electromagnetic waves, whichrequired an experimental verification, the demonstration of gravitational waveemission was accomplished in the early 1970s when R. Hulse and J. Taylordiscovered the compact neutron star (pulsar) binary system PSR 1915+21. Thisis a short period system that displays a systematic decrease in its orbital perioddespite the complete separation of the two orbiting bodies. The decrease in thebinding energy can be explained by the radiation of gravitational waves that causesthe system to contract.

Cosmology

Having completed the general theory in 1916, the following year Einstein followedthe same expositional path as Newton and extended the new conception of gravi-tation to the biggest problem of all, the structure of the universe. Around a pointsource, as Laplace had developed the Newtonian picture of the gravitational force,there is no tide for an orbiting point. An orbit, whatever its angular momentum,is always closed. But as we have seen, for a point in the relativistic case thereis a variation in the mass (inertia) of the orbiting body and therefore the angularmomentum is not constant although it is conserved. Thus, as is the case for anextended mass, the central body creates a tide that even a point mass in motionfeels. At the cosmological scale, the overall mass distribution, creates a local forcethat also contains the equivalent to a tidal force according to the Poisson equation.The divergence of the force, for a general distribution, causes this. The same mustbe true for a relativistic framework since the covariance principle allows us totransform from one reference frame to another. The curvature of an equipotentialsurface is another way of saying we have a tidal term. For a point source this isclear, as you get close to the central body any finite region of space experiences asecond derivative of the potential, the curvature is the inverse square of the lengthover which this gradient occurs and this is proportional to the density. Thus thetimescale for freefall becomes comparable to the light crossing time and at thatpoint we have an event horizon. Einstein’s intuition was that the tidal accelerationis the same as a curvature and if the geometry of spacetime is global the localcurvature produced by the large scale distribution of masses will yield the gravita-tional field of the universe and determine the motion of all constituent masses. Onan equipotential surface the motion is inertial. But if we have a tidal force, thereis a change in the gradient to this surface from one point to another, the surfacebehaves in space as though it is curved. The important difference between clas-sical and relativistic gravitation is that the spacetime, not simply space, changes.That means there is a possibility that two observers can see a differential redshiftof light, not from relative motion but from differences in the local density aroundeach, and the spacetime itself will seem to be expanding. This is the basis of


cosmology, the origin of the observed relation between distance and redshift. Togo into more detail, you should consult Craig Fraser’s book in this series.

IS THIS ALL THERE IS?

Usually in histories, we ignore anomalies and dead ends but some have beenso prominent, and so comparatively long-lived that I don’t think it appropriate tosimply pass over them silently. This brings us to the edge of precision measurementand also a fundamental problem in modern cosmology. While it is beyond the scopeof this book to review modern relativistic cosmology (again, I recommend the bookby Craig Fraser in this series for more discussion), I will mention the fundamentalproblem and its solution in a semi-classical framework to again demonstrate thepoint that both methods and questions persist in physics for far longer than theysometimes seem to.

As we saw, Newcomb proposed a modified gravitational force law as a solution tothe perihelion precession anomaly for Mercury. In the 1970s, the rotational motionof disk galaxies was shown to require a significant component of nonluminousmatter, now called Dark Matter, homogeneously distributed through the systemand extending to large distances from the center. In time, this has grown toabout 97% of the total gravitational mass. While the nature of this matter isheatedly debated, and likely will be unresolved even as you are reading this, thisparadoxical situation has spawned two proposals that bear striking similarities tothose predating General Relativity. The first proposal comes from cosmology andis directly linked to the explanation for the structure of the universe provided bygeneral relativity. You’ll recall, I hope, the discussion of the virial theorem, whichstates that the there is a fixed sum for the total kinetic and gravitational potentialenergies for a bound system. If we imagine a cluster of galaxies instead of a cloudof gas particles, bound to each other by mutual gravitational attraction, classicalNewtonian gravitation is sufficient to determine the total amount of binding mass.You measure the velocity of each galaxy and find the sum of the kinetic energies.Then, measuring the projected size of the ensemble, the radius of the cluster, youcan compute how much mass is required to keep the system bound. This can bedirectly compared with the masses of the individual galaxies obtained from theirstarlight and from observing the motion of gas bound within them. How this is donewould take us too far from the point. What has been found after several decadesof study, and this will certainly be ongoing even as you read this section, is thatthe mass required to keep the clusters together exceeds that of the visible galaxiesby up to ten times. Because this matter exerts only a gravitational force, henceis massive, but emits no light at any wavelength it is called “dark matter.” Itsdiscovery has been one of the strangest consequences of relativistic modificationsto gravitational theory because it is not only possible to include it in the descriptionof the expansion of the universe, it is actually required. How this affects our notionsof forces depends on what you think it is. There is also evidence that the universeis now expanding at an accelerating rate and this requires a repulsion in thespacetime now. No known force is capable of producing this. Gravitation, as youknow, is only attractive and this effect seems to be important only on very largescales, of the order of the size of the visible universe or about ten billion light


years, and not in local scale of the solar system, about ten light hours. If this iscorrect, it points to the most significant modification in our ideas of matter in thelast three hundred years.

NOTES

1. One way o see why this is necessarily a consequence of the invariant speed of lightis o remember that a speed combines the dimensions of space and time. If the time intervaldiffers between the two observers so must the length interval to compensate. But this alsomeans that as the speed approaches c the finite time interval of the comoving observerbecomes an infinitely long one for the stationary observer and vice versa.

2. Sommerfeld used this to explain the fine structure of spectral lines, assuming strictquantization of both the orbital angular momentum and this perturbation. In GRT this isnot necessary since the orbits form a continuum but the relativistic mass effect remains.

3. To be more precisely of the metric. Light follows a null geodesic, meaning thatin the language of our previous discussion d s2 = 0 so that the components are relatedby g00dt2 = g11dr 2 + r 2dω2 where dω includes all the angular terms for a sphericallysymmetric spacetime where the g coefficients depend only on the radial distance from thecenter of the mass. The angular deflection is a consequence of the potential at r and isnearly the same as the ratio of the distance of closest approach, b to the Schwarzschildradius, �θ = 4GM/bc2 = 2R�/b. For a solar mass, R� is about 3 km so this amounts toabout 1.8 arcsec at the solar limb, a very difficult measurement during a solar eclipse butnonetheless tractable even with the photographic emulsions and telescopes available atthe start of the twentieth century. The observational test, finding a value that within theuncertainties disagreed with classical theory and corresponded to Einstein’s prediction,catapulted Einstein to public fame as a virtual cult figure. Subsequent solar system testsincluded the direct detection of time dilation for signals propagating near the limb fromsatellites.


9

QUANTUM MECHANICS

There is no doubt that the formalism of quantum mechanics and its statistical inter-pretation are extremely successful in ordering and predicting physical experiences.But can our desire of understanding, our wish to explain things, be satisfied with atheory which is frankly and shamelessly statistical and indeterministic? Can we becontent with accepting chance, not cause, as the supreme law of the physical world?

—Max Born, Natural Philosophy of Cause and Chance (1949)

Atomic structure at the end of the nineteenth century was a mix of combinationalrules derived from experimental chemistry and simple physical models. The in-vestigations of the eighteenth and nineteenth centuries had been attempts towardthe microscale applications of force laws, combining mechanical and electro-static ideas but more directed toward the interactions between particles than theirstructure. These two different streams converged during the second half of thenineteenth century mainly through spectroscopic discoveries: the identification ofspecific unique sequences of emission lines that fingerprinted the elements.

THE ATOM AND THE ELECTRON

At mid-century, Kelvin and Helmholtz had demonstrated the persistence of vor-tices in a fluid in a celebrated theory dealing with their circulation. This quantityis the integral of the velocity around the circumference of a vortex ring or tube andKelvin’s theorem states that it is a conserved quantity in the absence of viscosityan a constant in an ideal fluid. It means that two vortex rings can pass through eachother, collide and bounce, and generally behave as though they are fundamental“elements” of a fluid. In the dynamical theory that resulted from this theorem,expounded most completely by Kelvin in his Baltimore Lectures (1884), counter-rotating vortices attract through a force of the form v × ω where the vorticity, ω,is defined as the circulation of the velocity field v, This is the Magnus force. A


kitchen experience illustrates the basic idea. Imagine using an egg beater to whipcream. The rotors turn in opposite directions, accelerating the fluid between them.The flow is described, then, by Bernoulli’s law—the pressure drops and the beatersattract. In a world filled with an incompressible ether, it’s easy to see how it is just ashort step out of the kitchen and into the atomic realm. The stability of vortex ringswas especially fascinating, so much so that Kelvin’s collaborator, Peter GuthrieTait, regularly used a smoke-ring generator in his demonstrations. The rings aremore intuitive objects than spherical vortices, having already been considered inCartesian mechanics. But Descartes did not possess any mechanism that couldrender such structures dynamically stable. Kelvin and others added the possi-bility for the vortices to vibrate in nonradial (that is, circumferential) modes thatnaturally produced harmonic series. Thus, if they were electrical bodies, thesepulsating vortices could produce electromagnetic emission at specific, uniquefrequencies—emission lines—although the mechanism is left unspecified. Theresearch program occupied a number of Cambridge physicists, among whom wasJohn Joseph Thomson (1856–1940) for whom the dynamics of vortex rings was thebasis of his 1884 Adams Prize essay. By the end of the century, albeit with fewadherents, this model appeared viable as an explanation of atomic phenomena.

In contrast, the electrical program led to an atomic model that consisted of pos-itive and negative charges homogeneously distributed and the mass coming fromself-interactions. Such a model, whose charging properties could be understoodby increases of deficits of one or the opposite charge, was especially successful inexplaining the chemical laws. It was an electrostatic, not a dynamical, explanation,an essentially structureless spherical mix of charges that insured neutrality. Itschief explanatory importance was the structure of materials, not spectra, which itcould not handle. But it could explain how the electrical properties of gases changewhen subjected to a sufficiently strong field. When a gas is ionized, it conducts.This change was explained by the liberation of free charge by the exciting agent,whether an electrostatic field, light, or a current passing through the gas. Theprogram of extending forces to the microstructure of matter was completed withthe discovery of the electron and understandings its role in the structure of theatom. The three fundamental demonstrations of the existence of this particle stillrequired classical fields. Two of these were similar to mechanical experiments.The third was more indirect, based on spectroscopy.

The first discovery was directly linked to the structure of the atom. Peter Zeeman(1865–1943) found that atomic spectral emission lines split into definite patternswhen a magnetic field is applied. He was inspired by a casual remark of Faraday’sabout the effects of the field on the structure of matter. Specifically, Zeeman foundtwo patterns. Along the direction of the field, there were always two of oppositesenses of circular polarization. Transversely, a third appeared between the othersand the lines were linearly polarized. This was immediately explained withinLorentz’s electron theory by the application of two forces, electrical and magnetic,through the inductive force. As long as the fields are static, the trajectory is ahelical path around the direction of the magnetic field line. Acceleration alongthe field is only due to an electric field. Now since light, as an electromagneticwave, has a phase for the oscillation, natural light is incoherent and is a mixture ofthe two senses of rotation: clockwise and counterclockwise. The electron, instead,

Quantum Mechanics 181

has a helicity depending on its charge if the magnetic field is static. So onesense of polarization will be resonant at the frequency of the orbit, the other willpass freely. In the orthogonal direction, one should see two lines separated by aquantity that depends only on the dynamical magnetic moment of the charge, andone that is the initial un-displaced line. Along the field direction, the two sensesof polarization produce opposite senses in the electron which is resonant at therelevant frequency. Thus the separation of the components varies linearly with themagnetic field strength, as Zeeman found, and has a pattern that depends only onthe orbital properties of the electron and its charge to mass ratio, e/m.

The second experiment used a remarkably simple vacuum apparatus calledthe Crookes tube, an evacuated long glass cylindrical container with en electrodeat one end and a phosphorescent screen at the other end that acted as a secondelectrode but also registered the impact of the current with a visible spot. J. J.Thomson employed this device to measure the properties of the particle responsiblefor electric current. His tube contained a pair of metallic plates to produce a localelectric field and was placed in a magnetic field whose force could be adjusted.Because the Lorentz force depends on the charge, from the electric and magneticfield, and the mass, the charge to mass ratio could be measured when the electricfield was changed and the magnetic field adjusted to cancel the deviation. Thevalue he found was almost 2,000 times larger than for the weight of the hydrogenatom (which, although neutral, had a known mass so its charge to mass ratio wasinferred). This was the electron, the first elementary particle to be identified. Toperform this experiment, notice that Thomson needed the force law for a current soa microscopic particle was discovered using a classical force law. It’s ironic thatThomson received an honor for his work on the vortex atom and then obliteratedits basis with the discovery of the electron about a decade later. This is still adaily experience for anyone who uses a standard television.1 The deflection ofthe electrons is the same but the pattern is more complicated. A color televisionscreen, rather than being a continuous film, is discretize into dots of various colorsand three beams are steered together with modulated intensity of the currentdepending on the input signal. The same technique was also the origin of themodern oscilloscope.

The third experiment used charged oil droplets, which are good conductorsand retain the charge, suspend them in an electric field against gravity underhigh vacuum. From the measurement of the size of the drops using a magnifierand seeing at what field they remained in equilibrium, the quantity of liberatedcharge could be determined. This ingenious experiment was performed by RobertMillikan (1868–1953), who also showed the discreteness of the charge in terms ofa fundamental value. While the Thomson experiment yielded the charge to massratio for the charges in the current, the Millikan experiment directly measured theunit of charge. The measurements of the charge of the electron, and its charge tomass ratio, were based on mechanics and magneto- and electrostatics. They showedthat the charge is equal to that of the ions, that a unit of charge corresponds to theatomic number already found by the chemists, but that the mass of the particle isextremely small, of order 0.1% that of the ions. This cannot account for the bulkof the mass of the atom. The discovery of charge as a particle property, along withelectrodynamics, completed the classical program of applying force to field.


THE STATISTICAL MECHANICAL ROOTS OF THEQUANTUM THEORY

Maxwell and Gibbs, in their development of statistical mechanics, had successfullydescribed how to treat the interactions between particles in a gas and, with that,to derive the gas laws. But they had been able to almost completely ignore theirinternal structure. In short, they had not addressed the structure of the atom.Boltzmann, who had been one of the principal contributors to the continuumtheory, also provided the first model for how the internal energy is distributed.He assumed that any atom is a bound system consisting of discrete energy levels,states, whose occupation was statistically determined only by the state’s energyrelative to the mean thermal energy of the system and on the statistical weight ofthe level—the number of ways the individual states can be populated.2 Spectrallines furnish the key since these are characteristic of the substance and at discretefrequencies. How the states are structured, and why they don’t form a continuum,were separate questions. Transitions between states were not explicitly dealt withnor was there any explanation for how the radiation and matter couple. However,that wasn’t necessary. Boltzmann was treating systems in thermal equilibrium.Since the rates for all individual processes collectively balance, it suffices to knowthe population distribution they generate, thus avoiding any “how” questions.An important feature of this approach, you’ll notice, is how it avoids any forces.The states are determined somehow; this was proposed before the discovery ofthe electron so Boltzmann didn’t even detail what he meant by the energy levelsthemselves. That would come later. It was enough that the energies are discrete.His distribution was the probability of a number of atoms in a gas being in somestate relative to the lowest energy level, the ground state, and because it was anequilibrium calculation didn’t treat the individual atoms, just an ensemble.

There was another connection between light and microphysics. Boltzmann hadfound the relation between the energy density of equilibrium radiation within aclosed, hot cavity and its pressure. The interaction with the walls was a separatematter. In equilibrium, all radiation incident on the wall is either absorbed orreflected. For a blackbody radiation, the limiting case, the energy is absorbedequally at all frequencies and is in energy balance between the radiation absorbedand emitted independent of the frequency, there is a spectral energy distributionin equilibrium whose total flux depends only on temperature. Kichhoff and Bunsenhad found, earlier in the nineteenth century, that the ratio of the emissivity to theabsorption is a universal function only of temperature but the precise form of thespectrum was unknown. Its integrated intensity had been measured in 1879 byJozef Stefan (1835–1893) and Boltzmann’s derivation of the radiation pressurelaw also predicted the measured T4 temperature dependence. There seemed tobe a connection between the atomic states and this radiative process since thecavity is at a fixed temperature and the radiation emitted by the walls is due to thede-excitation of the constituent atoms. As for Boltzmann’s atom, the distributionis formally universal, independent of the properties of the matter. This was theapproach adopted by Max Planck (1858–1947) who, in 1900, announced the lawfor the energy distribution of the radiation. He was thinking about the radiationitself, combining modes in a cavity and assuming the individual oscillators would


not interfere. The electromagnetic field was modeled as discrete packets of light,the quanta, whose energy was assumed to be an integer multiple of a fundamentalquantity, the frequency times a new physical constant, h that has the dimensionsof action (that is, energy times time or momentum times displacement). Theseoscillators only radiate by transitions between the stationary states, the energiesare not the same as those of the emitted photons and only the interaction pro-duces a change. The law agreed perfectly with experimental measurements of thespectrum of blackbody radiation but it left the nature of these quanta of actionunexplained. There was another connection between light and the atom. This wasthe photoelectric effect, the emission of charges when a metal cathode is exposedto blue or ultraviolet light, that was discovered by Philipp Lenard (1862–1947)in 1902. There were two parts to the effect. He used a high-intensity carbon arcto illuminate a metal plate in an evacuated tube, similar to that used by Thom-son, but with an anode whose potential could be changed to prevent the impactof low-energy electrons. He found that the intensity of the current was directlyproportional to the intensity of the light but not the energy of the electrons, whichdepended instead on the wavelength of the light. Millikan extended Lenard’s re-sults and found that the constant of proportionality had the same value as Planck’sconstant of radiation. The photoelectric law, that the energy of the ejected elec-tron is linearly proportional to the frequency of the light, was an unexplained butfundamental result.

This is where Einstein started in 1905. He was already thoroughly immersed inthinking about how to apply random processes to microscopic phenomena from hiswork on Brownian motion. Now he addressed the problem of how fluctuations wouldinteract with a bound system. His approach was to abstract atomic structure asfar as possible, reducing the system to a two level harmonic oscillator. FollowingPlanck, he assumed that emission and absorption take place only in discretejumps, which were subsequently called “photons.” I’ll now use that term to signifyPlanck’s quanta of action. These jumps occur with some probability distribution.He distinguished two types of transitions, those that were “stimulated” by theradiation in the cavity, the rate of which are proportional to the intensity, andthose that occur “spontaneously.” While the stimulated processes can either addor subtract energy from the system by, respectively, absorption or emission, thespontaneous transitions can only produce emission. This was also consistent withhis understanding of entropy in thermal baths. To see how this affects the radiation,we suppose an atom, with only two levels with energies E1 and E2 > E1, is insidea box within which there is a collection of particles with which it collides. Thereis also radiation in the box, it’s hot, and the radiation is both absorbed and re-emitted by the atom. Imagine that we have not one but a slew of atoms, and therelative level populations are given by the Boltzmann distribution because theparticle collision time is sufficiently short relative to any radiation absorption oremission process. We now compare the rates for only the radiative processes. Thatwhich produces absorption of a number of incident photons, with arrival rate Iat the frequency corresponding to the separation of the levels, can be absorbeddepending on the relative population of the lower level and the cross section forthe interaction. Einstein wrote this as (rate of absorption) = n1 B(1 → 2)I . Heren1 is the number of atoms in the first state relative to the total number in the


box, the “population,” and I is the incident or emitted intensity. He assumed thetwo electromagnetic distributions would be identical in equilibrium. The inverseprocess, when a photon produces a downward transition, is called the stimulatedrate, (rate of stimulated emission) = n2 B(2 → 1)I since it is assumed the same“type” of photon can produce a de-excitation of the upper level whose populationis n2. The B coefficients are the probabilities of the process (actually, they’re therates but the assumption was the processes are random in time).

If this were the only set of possible processes, the balance between themwould depend only on the relative transition probabilities and not the intensityof the radiation. But when the upper state is excited, it can spontaneously decay,independent of the energy density of the radiation, with a rate n2 A(2 → 1), A beingthe probability of a spontaneous emission in an analogous manner to B. This is astatistical process, one produced by the fluctuations in the level populations andthe chance that one of these will be large enough to cause the system to relax. Itis inherently unpredictable. Therefore, there is a source for new photons, one thatis independent of the radiation and only comes from the populations themselves.Einstein’s next step was to connect the radiation with the mechanical processes inthe gas, to say the level populations are produced only because of the collisions,the rate of which depends only on the temperature, as Boltzmann had found. Theratio of the level populations then depends only on their energy difference. Theradiation density and spectral distribution in the box will be precisely what arerequired to maintain the equilibrium between the radiation and the oscillatorsin the box. This was Planck’s model and Einstein showed that if the transitionprobabilities are constant he recovered the known result for blackbody radiation.To do this, however, Einstein made a revolutionary assumption, that the energy ofthe radiation of frequency ν is �E = hν and corresponds to the energy differenceof the two levels. This was the step that Planck hadn’t taken. It explained thephotoelectric effect and it is the “origin” of the photon. It also followed that theradiation will be independent of the level distribution of the atomic system and,because only ratios of the probabilities and ratios of the level populations arerequired there is a direct relation between the probability of absorption B(1 → 2)and emission B(2 → 1), and that of spontaneous emission A(2 → 1) that holdsfor every pair of levels in the atom. The almost astonishing simplicity of thisargument may hide the essential conundrum. This is a relation between statisticalquantities, not deterministic ones. The absorption and emission occur only withrelative probabilities, not deterministic rates. The Planck distribution is then,as was the Maxwell–Boltzmann distribution, the relative chance of observing aphoton at a given energy, not the certainty of seeing it. A purely deterministicmechanical process might produce the radiation incident on the cavity but theradiation inside is completely random.

Atomic Structure and Quantum Theory

Radioactive emission provided a new tool to probe atomic structure. This consistedof using α particles, the least penetrating of the three types emitted by radioactivesubstances. These carry positive charge and have the mass of a helium atom.They provided Ernest Rutherford (1871–1937) the tool for probing the structure


Figure 9.1: Niels Bohr. Image copyright History of ScienceCollections, University of Oklahoma Libraries.

of atoms in solids, where their fixedarrangement simplified the dynamics.The scattering experiments he per-formed in 1911 established that themass of the nucleus is centrallyvery concentrated. The experiment wasbeautifully simple. A beam of α par-ticles was aimed at a thin metal foil;Rutherford chose gold because it has alarge atomic weight, hence a large posi-tively charged constituent, and could beeasily milled into very thin layers. Theparticles scattered in an unexpectedway. Instead of a broad range of scat-tering angles, that was expected expectfrom a uniform distribution of charge,most simply passed directly through thefoil with minimal deviation and onlya very few backscattered. Rutherfordshowed that this could be explainedusing a classical two body calculationin which like charges interact electro-statically, one in a stationary pointlikemass and the other, lighter charge mov-ing in hyperbolic paths. The motion wasinconsistent with a uniformly mixed,neutral blob. Instead, the scattering ex-periments required the bound atomicelectrons to be far from the nucleus.Their location would be explained if they execute bound orbits around the centralcharge.

This solar system in miniature was the starting point for Niels Bohr’s (1885–1962) idea of quantization. Bohr was a research associate with Rutherford atManchester around the time the new atomic model was being formulated. Inessence, it is a trick, a way of accounting for the stability of the orbits while at thesame time permitting emission when necessary. The problem came from classicalelectromagnetic electron theory. A moving charge radiates and, since it’s losingenergy, its orbit decays continually. This would lead to an almost instantaneouscollapse of matter which, as your daily existence demonstrates, doesn’t happen.Instead, Bohr assumed that the orbital angular momentum is assumed to take onlyspecific, discrete values. A proportionality constant with the right dimensions wasalready known for this, h, Planck’s constant. He also assumed the photon picture fortransitions between these levels, that they occur according to Einstein’s statisticalformulation of the photon, and that the change in the energy produced the emissionof a pulse of light, a photon, whose frequency is given by Einstein’s photoelectricrelation �E = hν. The force maintaining the orbits was classical, electrostaticattraction of the nucleus, and the electrons were treated as pointlike orbiting


low-mass particles. The change from the classical dynamics was the introductionof a quantization rule for the levels. Bohr then assumed that the discreteness of theorbital angular momentum corresponded to a discrete radial distribution in whichthe distance from the nucleus is an integer multiple, n of a fundamental length, a0,now called the “Bohr radius” (about 0.5 Å or 5 × 10−5 microns) and obtained theknown hydrogen line series discovered by Balmer and Lyman and predicted theproportionality constant for the frequencies that had been determined by Rydbergin terms of the charge and mass of the electron, e and me, and h with no freeparameters. His next success was the prediction of the spectrum of ionized heliumfor which the reduced mass of the orbit shifts the center of mass because of thedifference in atomic weight between hydrogen and the one electron helium ion.The series, discovered in the 1890s by Edward Pickering in the optical spectraof very hot stars and at the time unobtainable in the laboratory was a tremendoussuccess of the model. Laboratory line spectra thus became the tool to delineatethe permitted orbits within this system and atomic structure could now be studiedby proxy.

The fine structure of orbits was explained by a separate but related mechanism.As it did for the Newtonian problem of gravitation, the Kepler problem appearedin the atom but now in a different guise. Recall that the rest mass is not theimportant thing for a moving mass, its inertia depends on its motion. Thus, for anatomic orbit, the mass of the electron depends on the local motion. If the angularmomentum can be distributed discretely for any energy in units of h= h/2π ,the eccentricity of an orbit is determined by the ratio of the binding energy to theangular kinetic energy through L, the total angular momentum. If we think of theorbits as ellipses, in a central potential they close. But if there are other charges,or if the mass of the electron isn’t constant, the orbit will precess. If this newmotion is also quantized, the amplitude in energy—the splitting of the lines—willbe smaller than that produced by the angular momentum but it will be a necessaryeffect of the dynamics. This explanation, due to Arnold Sommerfeld (1868–1951),not only appeared successful but furthered the program of modifying the classicalnotion of force to the atom.

The problem of how these orbits arrange themselves stably was a differentstory. Bohr postulated that they group in shells with a maximum permissiblenumber. This explained the origin of the statistical weights for the levels thatBoltzmann had introduced as the irreducible number of possible combinationsfor particles in each atomic state, independent of temperature. But there was noexplanation for how this arrangement was achieved. The clue came from helium,the simplest two electron system. Wolfgang Pauli (1900–1958) introduced the“exclusion principle” (1925) to explain the absence of some transitions predictedpurely on the basis of the distribution of the atomic levels, by stating that no twoelectrons can occupy precisely the same state. This required introducing another,unexplained, quantum number to those required for the radial and angular motions.In an attempt to explain both the Zeeman effect and the Pauli principle, in the sameyear Samuel Goudschmit and George Uhlenbeck proposed that spin would accountfor a contribution to the magnetic reaction of the atom by endowing the electronwith its own intrinsic magnetic moment. The effect of an imposed magnetic fieldtherefore aligns these spins and splits the atomic levels to produce the Zeeman


effect for the subsequent transitions. The difficulty was immediately pointed outby Lorentz, that since the electron is a pointlike mass it could not have an axialrotation. Nor was it clear why the number of components should be half-integermultiples of the Bohr magneton, the elementary unit of magnetization. But despitethese problems, Goudschmidt and Uhlenbeck’s paper was sent to Naturwissenshaftby their professor, Paul Ehrenfest, and subsequently an English version waspublished in Nature accompanied by a recommendation from Bohr.3 Thus wasthe spirit of the times. Even crazy proposals had a chance of being correct. Theexplanation for the strange factor of 1/2 came surprisingly quickly. The dynamicalproblem of an orbiting, spinning mass was solved by L. H. Thomas (1926) inanother application of relativistic mechanics to atomic structure, the motion of aspinning particle orbiting in an electrostatic field. He found a multiplicative factorof 1/2 that appeared to resolve the last quantum mystery. The solution later becamethe basis of the fully coherent theory of synchrotron radiation, the emission of lightdue to free motion of electrons in a magnetic field by Julian Schwinger but, as wewill see, with the demise of the “mini-solar system” model the explanation isn’tapplicable to atomic structure.

Bohr was also able to treat molecular motion within the quantum theory byassuming the quantization rule for each available motion. Since molecules arelinked atoms, and therefore neither spherical nor pointlike, the nuclei were treatedas masses glued together by the orbiting electrons. This is actually a very strangerecipe, it combined multi-center dynamics (which we saw from the three bodyproblem leads to incommensurable orbits and ultimately to chaos) and a numberof paradoxes of motion, but it preserves the force concept. Atoms in moleculesbehave as if tied together by springs, hence the interaction potential is imaginedas a source for a spring constant from which the oscillation frequencies can becomputed. For rotation, the simplest procedure was followed, to determine themoment of inertia for any geometry of the component atoms and then to quantizethe rotational motions in the same manner as the orbits. The regularity of theindividual transitions was accommodated by writing EJ = 1

2I J (J + 1) h where Iis the moment of inertia and J is the rotational angular momentum, and to thenuse the Einstein–Bohr relation for the difference between energies of the rotatorEJ ′ − EJ = h

2I [J ′(J ′ + 1) − J (J + 1)] to find the frequency. The differencesbetween the transitions consequently form a ladder. The same idea allowed thevibrational levels to become Ev′ − Ev = hω0(v′ − v) where ω0 is the resonancefrequency and v and v′ are the vibrational quantum numbers. Thus, by 1925,quantum theory had been successful was essentially a classical analogy with anumber of apparently ad hoc rules for assigning states to the orbiting particles. Itleft unanswered, however, more basic questions of stability and dynamics.

QUANTIZING DYNAMICS: A NEW MECHANICS

In the quantum theoretical atom, neither the transition processes nor the relativestrengths of the emission lines were explained. Bohr was treating only structure,how the electrons arrange themselves in the binding electrostatic field of thenucleus. As such, making the assumption that the states remain stationary andthat each periodic (bound) motion is quantized, the model provides an explanation


for the discrete levels and their relative statistical weights (their degeneracies)while still avoiding the details of the transitions. In effect, the picture is again anequilibrium, of sorts: if a rule is imposed that forces the states to be stationary,that the orbiting electron cannot radiate, then by imposing a discretization in thedistances and angular momenta you obtain the spectral sequences (in the oneelectron case) and a method for inverting the observed line distributions to obtainthe quantum numbers for the derived states. But why the atomic states shouldbe quantized was not addressed, or indeed addressable, within the theory. Thatrequired something more, a new mechanics. From the beginning, the importantdifference between the classical and quantum pictures is that h, the Planckconstant, has the dimensions of action and angular momentum. Although both areknown from macroscopic dynamics, they are continuous properties of a systemand neither has a characteristic value.

If on the scale of everyday life masses and charges follow the classical laws ofmotion, or their relativistic extensions, there must be a way to connect this with theatomic realm. How to make the transition was outlined by Bohr in his third paperon atomic structure in 1913. In what he called the “Correspondence Principle,”he asserted that the way from classical mechanics to a microscopic system was toquantize the action in units of the Planck constant, h, and to make all derivativesdiscrete by taking the change in the energy between states. It was, in effect,going backwards from a smooth to discontinuous rate of change of any measurablequantity, contrary to the way that the calculus had originally been developed! ForBohr, this was plausible because the existence of spectral lines and energy levelsdemanded that a dynamical quantity couldn’t ever change by an infinitesimalamount, these had to be discrete jumps. This required a complete change inthe Newtonian idea of a force. Accelerations became impulsive changes in themomentum per unit mass, similar to collisions. Time derivatives were replacedby multiples of the frequency and, you’ll recall, the frequency is a change in theenergy when going between quantum states. In Hamilton’s formalism of mechanicsthe frequency is the change in the Hamiltonian, or total energy, with respect to theaction. But now, since the action is quantized, as is the energy, the photoelectricrelation of frequency and change in the energy was recovered.

By the beginning of the 1920s, quantum theory was a mass of tools, guesses,and analogies in search of a foundation. Guided by Bohr’s axioms, the continuousdynamical equations could be quantized by taking any changes with respect tothe action to be discrete. But the difficulty was that although the orbital pictureexplains spectra, it fails to “save the phenomena” for matter. Building atomsis simple, a problem in discretizing the manifold of possible orbits among theinteracting electrons in the electrostatic potential of the nucleus. But molecules,and more complex aggregates, were much more difficult to imagine. The orbits ofthe individual atoms must somehow overlap and link the constituents to form stablestructures, an impossible task in so simple a system as the hydrogen molecule(although not for the ionized molecule which is a three body problem). The demiseof this micro-celestial mechanics all but severed the last direct connection withthe classical mechanical roots of the microscopic picture. The one model thatremained was the harmonic oscillator and the principal tool was the Hamiltonian.By analogy with the Hamiltonian construction of mechanics, the change of the


energy with respect to the action is the frequency. Then it follows, by constructionof the quantized system, that the difference in the energy between two levels isthe energy of the emitted quantum so any changes in the system the time alsosomething occurs by jumps.

Wave Mechanics

Wave mechanics, the first step in the transformation of the force concept in themicrophysics, was initiated by Louis de Broglie (1875–1960) when he showed howa particle can be treated like a wave, an idea suggested by the comparison of theclassical and quantized picture for electromagnetic emission, in which a mass mhaving momentum p appears to behave like a wave with a wavelength λ = h/p.At about the same time, Arthur Compton (1892–1962) had sought a descriptionof photon scattering by free photons assuming a nonrelativistic particle picture.Let’s digress for a moment to see how that works.

A parcel of light carries a momentum p = E/c = hω/c. If an electron at rest ishit by a photon, the light scatters. Since the momentum of the photon is along itspropagation direction, assuming the scattering takes place at an arbitrary angle,the electron gains energy at the photon’s expense. Energy and momentum areconserved in this collision. But, and this is the most important point, a change inthe photon’s momentum (and energy) means its frequency also changes: increasingthe momentum of the electron produces a downshift in the photon’s frequencybecause the parcel of light is massless and, as observed by Compton in 1922, X-ray photons lose energy while the electrons gain momentum. A new feature is thatthe collision has a threshold at characteristic wavelength that depends only on theparticle’s rest mass, λC = h/(m0c), that is now called the Compton wavelength.Alternatively, we can say the process is nearly resonant when the photon energyis equal to the particle’s rest energy, so the governing parameter is hω/(m0c2)which, when equal to unity, gives the Compton wavelength. The advantage of thisapproach is how it avoids any questions about how the interaction takes place.By using a simple ballistic collision, the course of the particle and photon can betraced to and from the collision without introducing any model for the interaction.By this straightforward calculation, made revolutionary by its explicitly includingphotons as “particles of light,” a striking result had emerged: the energy of a photonchanges after it hits a stationary electron but the change depends on the anglebetween the incident and final directions of the scattered photon. The electron isaccelerated and the energy and momentum behave for this system precisely as ifthe photon is a relativistic particle with momentum but without a rest mass.

A similar picture was exploited by de Broglie in his doctoral thesis (1923–1924)at Paris under the direction of Langevin, although he knew neither Compton’s re-sult nor its explanation. De Broglie knew that light is classically well describedas a wave phenomenon. The controversy of the first part of the nineteenth centuryabout whether it is a particle or wave had long before been, apparently resolvedexperimentally and theoretically in favor of the wave picture. The success of thephoton changed this consensus. Instead of forcing a choice, de Broglie acceptedthis duality and asked what would be the consequence of extending this to mas-sive particles, e.g., the electron. Within the Bohr atom, although exploiting the


quantum numerology to derive orbits, the origin of the stationarity of the stateswas unexplained; it was imposed by construction. If instead the electron could bewavelike, this discreteness could be naturally explained as a standing wave. Atcertain, fixed distances from the nucleus a bound electron could, if it is a wave,interfere unless the state is a harmonic. Thus it would be a natural explanationfor the structure of matter. The characteristic wavelength he computed dependsonly on the particle’s momentum, p, λB = h/p. Since the momentum of a boundorbiting electron is related to its energy by the virial theorem, this yielded thecorresponding states. The success of this picture, along with a relativistic justifi-cation contained in the thesis and the first of a series of papers resulting from it,made this a serious contending model for matter, albeit a very strange one. It had,however, some particularly attractive features. The model naturally exploited thephase space of a particle. Recall that h has the dimension of action and that h3 isthe same as an elementary volume of phase space. Thus, you could say, if there isonly one particles per unit in phase space, for some reason, then you would expecteach to have an equivalent “width” of the de Broglie wavelength for a momentum.

Langevin sent a copy of de Broglie’s thesis to Peter Debye (1884–1966) at theETH in Zurich and to Einstein at Berlin. Both read it with interest. Einstein’sreaction was supportive and spurred further interest in the idea both at Paris andelsewhere, such was the importance of his imprimatur. In Zurich Debye asked hiscolleague, Erwin Schrodinger (1887–1961), to take a look at the work and reporton it at the group seminar a few weeks later. This turned out to be an auspiciousrequest. Although at first reluctant to consider it, Schrodinger accepted the task andin 1926 published his solution to the hydrogen atom and its generalization to atomicmechanics. He examined how this wave picture is indeed applicable to particlemotion in a bound system by using the Hamiltonian function. He used similarreasoning to the approach Hamilton had introduced with his eikonal function.4

Schrodinger realized that he could write a wave function whose exponent is theeikonal, or action, and that this would furnish the equation for a propagating wave.At first he tried a relativistic formalism and derived a second order differentialequation of a wave type with a potential, the precise form having already beenexamined by Helmholtz in the context of acoustics. But for the nonrelativistic casethe situation was different. The resulting equation is a diffusion type, first order intime and second order in space with the energy as a proper value, or eigenvalue, ofthe stationary states. For a freely moving particle, this yields a gaussian function.For a stationary particle in an electrostatic field, produced for instance by thenucleus, the eigenvalue spectrum is the same as the distribution of states in thehydrogen atom: the de Broglie analogy yielded the Bohr states.

The technical mathematical problem of how to solve the type of differentialequation of which Schrodinger’s is an example had been well studied for nearly acentury, starting with Liouville in the middle of the nineteenth century. Generalsolutions were known for diffusion and wave equations with both initial conditionsand fixed boundaries and Schrodinger was thoroughly familiar with that work.The equation he derived had, however, a special feature. The energy of the stateis the solution and, depending on the coordinate system, takes specific valuesthat depend on integers. In other words, the solutions are quantized and thiswas the most remarkable feature of the equation. The solution for the hydrogen


atom, a centrally symmetric electrostatic potential with a single electron, separatesinto three components, one radial and two angular. The solutions to the angularequations are a classes of function found by Legendre, while those in radiusare functions that had been found by Hermite, both long before Schrodinger. Toobtain the angle-dependent solutions he identified the integer with the angularmomentum, for the radial it was with the principle quantum number. In so doing,he recovered the dependence of the energy on the two quantum numbers thathad been discovered from the Bohr atom and spectroscopic analysis. It was nowpossible to compute atomic structure without orbits. But at the same time it wasfar from clear what these wave functions meant because they were not the thingmeasured, being instead the means for computing the energies by taking averagesover the volume. By the correspondence principle, any average is the same as theclassically measurable quantity (for instance, just as the mean of the Hamiltonian,〈H 〉, is the measurable energy, the mean momentum, 〈p〉 , is the measured value forthe distribution and that corresponds to the classical momentum). The dynamicalequations of classical mechanics could be recovered, although it required a changein the normalization to remove Planck’s constant (which is the single criticaldeparture from the classical result). Perturbations could also be handled, and thechange in the energies of atomic levels when an external electric (Stark effect) ormagnetic (Zeeman effect) is present could be computed using the same formalism.Also coincidences between levels, the degeneracy or statistical weight, followednaturally from the wave function representation. The machinery developed quicklyin Schrodinger’s hands, the entire foundation was laid within one year.

From Matrix Mechanics to Quantum Mechanics

At the same time, an alternative view of atomic dynamics was being developedin the Gottingen circle centered around the physicist Max Born (1882–1970)and the mathematician David Hilbert (1862–1943). Their approach was moreclosely linked to the Bohr program of quantizing dynamics of particles and or-bits by using operator methods. This was a route taken by Werner Heisenberg(1901–1976). Although his thesis work with Sommerfeld was on fluid turbulence,he was drawn to microphysical problems and on Sommerfeld’s recommendationjoined the Gottingen group. His starting point, unlike de Broglie, was to make theCorrespondence Principle the central axiom of an effort to discretize mechanics.In his paper of 1925, “Quantum-mechanical re-interpretation of kinematic andmechanical relations,” Heisenberg replaced the orbits with a harmonic oscillatorand wrote the change in the momentum according to Bohr’s principle, replacingthe position as a function of time with a set of discrete, quantized, amplitudesof oscillations whose frequencies depend on the change in the quantum number.By this substitution, he was able to solve for the energy of the oscillator in termsof the quantum state which he found to be an integer multiple of h times thefundamental frequency. But he was faced with another very strange result for theposition and momentum. For a classical phase space it doesn’t matter if we takethe produce xp or px but Heisenberg found that xp − px = i h, with i = √ = 1.The two variables didn’t commute. Although such algebras had been known sinceGrassmann and Hamilton—indeed, they had been the point of departure for


vectors and quaternions—nothing like this was expected from a mechanical sys-tem of operators. Heisenberg went on to show that the energy has a nonvanishingoffset; even in its ground state such a simple system would never have zero energy.In a last step, he showed that this could be extended to a nonharmonic oscilla-tor, one with a small cubic term that mimicked a perturbation. He immediatelypresented the result to Born, who recognized it as the algebra of noncommutingmatrices. Together with Pascual Jordan (1902–1980), Born derived the equationsof motion and in less than three months after the submission of Heisenberg’s paperthey published “On Quantum Mechanics.” The third paper in this series, connect-ing with atomic structure, was published in 1926 along with Heisenberg (oftencalled the “Dreimannerarbeit” or “Three Men’s Work”) in which they derived theequations of motion for a generalized potential law and quantum mechanics wasborn.

Why a harmonic oscillator furnishes the basic model can be understood bythinking again about orbits. An circular orbit has a minimum angular momentumfor a given energy so an increase in the angular momentum produces an ellipticalorbit. The particle oscillates radially in the potential field around the mean distancegiven by its binding energy. Thus, this looks like a radial periodic oscillationexcited with respect to a reference circular orbit that is the minimum state. Theoscillation has a characteristic frequency that depends on the potential, hence theparticle’s kinetic energy, because by the virial theorem the kinetic energy is strictlytied to the potential energy for a bound system. A departure from a simple 1/rpotential leads to anharmonic behavior which explains why Heisenberg treated thecubic perturbation. The classical result of Jacobi and Clausius transforms into thisnew framework directly by construction. If we now “quantize the motions”—thatis, we assume both the commutation laws and assume all angular momenta areinteger multiples of h—then the oscillators are stable (they don’t radiate, againby construction) and their energies are fixed only by the combination of theirquantum numbers. Thus we arrive at a picture of oscillators similar to that use byPlanck in his derivation of the radiation law but with very different consequences.Heisenberg used this to show that the spectrum is a linear relation betweenthe energy and the principal quantum number. Imagine a mass painted with aphosphorescent coating attached to a spring. Let the mass oscillate and, in a darkroom, take a time exposure of the motion. The image will show the ends as moreintense since the motion is slower at the turning points so the times are unequal. Ifpulled with a greater force the maxima are displaced depending on the energy (theextension) of the spring). Now for a sufficient extension the oscillation becomesanharmonic. This is the same as a probabilistic model: if we ask where there is achance, on casual observation, of finding the mass it will generally be closest tothe end. This looks just like an orbit where if we take a photo we will see the sameprojected image.

The resolution of the mysterious commutation relation at the base of the newmechanics came again from Heisenberg in 1926. He showed that this non-commutativity comes from how we know the phase space coordinates of a particle.He showed that there is an intrinsic, irreducible fuzziness to any particle. Inwhat he called the “uncertainty principle,” the more accurately you know one ofthe phase space coordinates the less accurately you know the other, showing the


�x�p = h. These are the conjugate variables from dynamics and by changingenergy and time he found that the result could also be extended to �E�t = h.This meant that if you have a state with a finite lifetime, for any reason, the shorterthe lifetime the broader the energy level. It also applied to particle motion. Italso accounted for particle decay since the rest energy is then inversely relatedto the decay time. This meant that a fluctuation in any of the conjugate variablesdetermines that in the other. For the momentum, this naturally explained whya particle could have a jitter. But this also provided at least a clue to solvingthe first conceptual problem, that of the jumps. Because of fluctuations in itsenergy, a stationary state also has a finite lifetime. The increased widths in thelines arising from successively higher transitions in hydrogen that gave rise tothe original orbital designations could now be explained.5 The epistemologicalprice seemed to be high, it isn’t possible to state precisely, with perfect accuracy,where a particle is if you have a measurement of its momentum. Heisenberg out-lined a thought experiment in a series of lectures in 1926 at Chicago that provideda way to look at this. He imagined a microscope (with γ -rays as the illuminatingphotons to insure the Compton effect) such that any measurement made on theposition of the particle inevitably and unpredictably disturbs its state of motion.Further elaborated by Bohr, this became known as the Copenhagen interpretation.

For the concept of forces, this removed the last support for the notion of absolutespace and time. For macroscopic bodies this uncertainty doesn’t matter, that’s whyNewtonian and relativistic mechanics work for things on a large scale. But whenit comes to how matter behaves at the level that requires quantum mechanics,the only statements we can make are about probabilities of things happening:only the distribution of values can be predicted for measurements, not specificvalues. Bohr added an interpretative framework to help resolve this confusion.In his Como lecture of 1927, he proposed a sort of philosophical compromise,the principle of “complementarity.” He proposed that since the two treatments ofquantum mechanics were compatible and were formally equivalent, as Schrodingerhad shown, the contrasting picture of particles and waves should be acceptedas working hypotheses. Much as Newton had proposed for gravitation, withoutknowing the “why” the mechanical predictions were sufficient to proceed alongparallel lines. Internal atomic dynamics was replaced with jumps and stationarystates with transitions being time dependent interactions through fields. Dynamicswould be represented by wave packets with the mass and charge of a particlea measurable quantity but accepting the intrinsic uncertainty connected withthe disturbance produced in its phase space position because of uncontrollableinteractions between the measuring device and the thing being measured.

An even closer connection with statistical representations came from a differentdirection. Shortly after the publication of Schrodinger’s first papers, including thetime dependence, Born realized the link between the wave function and probabil-ity theory and through this, introduced a fundamentally new interpretation of thequantum theoretical picture. From the time dependent Schrodinger equation, hefound that the square of the wave function had the same properties as a probabilitydensity in space, the chance to finding a particle at some point. This moves witha center of mass momentum 〈p〉 and obeys the continuity equation. He thereforeattributed this to the intrinsic “fuzziness” of the particle. This distribution follows


immediately from the wave function but also carries an important feature. Theamplitudes of two waves, in classical physics, interfere when superimposed. InMaxwell’s ether, this finds a natural analog in the superposition of two motions forthe elastic medium. In the quantum mechanical picture, this requires a changein the description. Instead we talk about correlations. But more to the point, twoparticles can now appear to interfere with each other! Each amplitude has anassociated phase and, when we co-add particles, we are adding them coherentlyin phase. The motion depends on the potential, the particle itself only statisticallyresponds to a “force.” This was highlighted by a thought experiment by Aharanovand Bohm (1959) that has since been verified experimentally. A beam of electronsis divided equally into two parallel paths that pass a current carrying solenoid.The magnetic field is confined within the coil, there is no gradient in the externalspace. But the two beams pass on opposite sides of the coil and, therefore, expe-riences phase shifts of equal magnitude and opposite sign. When they recombinethey interfere and produce fringes where the electrons are either correlated oranticorrelated despite the absence of a force. The potential alone is sufficient toaccount for quantum mechanical measurements.

At around the same time, at Cambridge, Paul A. M. Dirac (1902–1984) had beentold about Heisenberg’s work by R. H. Fowler, one of the few English physicistswho had both followed and understood the continental developments. Thinkingabout the commutation relations, Dirac made an important connection, that theywere analogous to a mathematical device known from Hamiltonian mechanics.When transforming between coordinate representations, called a contact trans-formation, the change is expressed using the Poisson bracket, which had alsoappeared in Lie’s transformation theory (see the appendix for a brief discussion).It’s a striking example of the generality of the canonical coordinate representationthat the same basic equations appear in such diverse cases. They were most famil-iar from celestial mechanics where they had extensively used to change betweenspatial coordinates to the more convenient, and more easily measured, orbitalparameters. For Dirac, the commutation rules became the fundamental axioms ofthe new mechanics, much as vectors and quaternions had almost a century earlier.He distinguished two types of numbers, so-called c-numbers, quantities that com-mute (classical), and q-numbers, those (quantum) that do not. He realized that thecanonical coordinate pairs were q-numbers while the measurables, eigenvalues,must be c-numbers. He also introduced a beautifully simple, general notation forthe states, the bracket, that at once corresponds to the statistical averaging overstates and expresses the properties of the operators. A ket, |n〉, is the state, a bra isits complex conjugate, 〈n|, and the expectation value is then 〈n|Q|n〉 for any Her-mitian operator Q.6 A similar notation had been used for the scalar product of twovectors 〈u, v〉 where it is commutative, i.e., 〈u, v〉 = 〈v, u〉 for real (u, v); it hadalso been used for Fourier transforms, with which one passes between canonicallyconjugate quantities, and for algebraic operations on generalized functions, Thetwo quantities are called orthogonal when 〈u, v〉 = 0 so similarly Dirac imposedthe rule that two uncorrelated states have 〈u|v〉 = 0 (Hilbert and other mathe-maticians had already used this sort of notation for taking scalar products, so itsextension was natural to quantum operations).


Dirac extended the concept of “state.” By analogy with photons, he took theprinciple of superposition as his central postulate along with the inherent indeter-minacy of measurements. A state can be decomposed into a linear superpositionof other states, the correlations of which are the scalar products or, as Diracwrote:

The intermediate character of the state formed by superposition thus expresses itselfthrough the probability of a particular result for an observation being intermediatebetween the corresponding probabilities for the original states.

In his The Principles of Quantum Mechanics (first published in 1930 and stillin print today) Dirac defined a fundamental shift in viewpoint in contrasting thealgebraic and continuous approaches, writing:

The assumption of superposition relationships between the states leads to a math-ematical theory in which the equations that define are linear in the unknowns.In consequence of this, people have tried to establish analogies with systems inclassical mechanics, such as vibrating strings or membranes, which are governedby linear equations and for which, therefore, a superposition principle holds. Suchanalogies have led to the name ‘Wave Mechanics’ being sometimes given to quantummechanics. It is important to remember, however, that the superposition that occursin quantum mechanics is of an essentially different nature from any occurring inclassical theory, as is shown by the fact that the quantum superposition principledemands indeterminacy in the results of observations in order to be capable of asensible physical interpretation. The analogies are likely to be misleading.

He realized that in Hamiltonian mechanics there is a special relationship be-tween the dynamical variables, the Poisson bracket (used extensively for coor-dinate transformations in celestial mechanics).7 Dirac then showed through ma-nipulation of the brackets that for any arbitrary pair of quantities, uv − vu =(constant) × [u, v] and therefore was able to arrive at Heisenberg’s relationsqp − pq = i h, not in terms of derivatives but the operators themselves, and iden-tified the constant factor as an imaginary number. This last result he called thefundamental quantum conditions. He then introduced a way of speaking that hassince come to be dominant in the description of quantum mechanics, that when hbecomes vanishingly small we recover classical mechanics. This is, perhaps, oneof the strangest features of the theory because it is saying, instead, that when thescale of the system becomes incomparably large with respect to the elementaryunit of quantum phase space (when V /h3 � 1, we return to the classical worldof macroscopic bodies. An amazing feature of this algebraic approach, aside fromthe fact that it worked, was its generality. The Gottingen formalism required con-siderably more manipulation of symbolic quantities and could not immediatelyconnect the operators with observables. Instead, by using the Poisson bracket asthe central operator, Dirac could immediately establish a broad range of dynamicalrelations. Most important, perhaps, was his realization that because the Poissonbracket in the quantum theory has the same dimensions as angular momentum—always the same ambiguity in dimension with the action—it was possible to extend


the commutation relations to something that looked like angular momentum as anoperation without the need to think about orbits.

Since the Bohr approach had been successful for molecular spectra it wasobligatory that the new dynamics should also work. A diatomic molecule, or anymolecule consisting of as many atoms as you would like, has several modes thatdon’t require any fundamental change in the idea of an elastic force law that meterlytakes discrete values for the energy. The vibrational modes of, say, a diatomicmolecule have the same frequencies as those given for any harmonic oscillatorbound by any potential; under small displacement the frequency is given by thesecond derivative of the potential (the rate of change of the force with displacement)and is a linear multiple of the vibrational quantum number, v, Ev = hω0(v + 1

2 );the fundamental frequency is ω0, much as would hold for a spring. The extra factoris the zero point energy, the same one found by Heisenberg. The rotating moleculeis almost as simple but lacks the zero point shift; its moments of inertia, I , areidentical around two axes vanishes along the internuclear axis, so if the angularmomentum J is quantized by a discrete number j , such that J 2 = j ( j + 1) hthen for each rotational state has an energy EJ = j ( j + 1) h/I .8

The Least Action Principle and Sums over Paths

The last reformulation for quantum mechanics, by Richard Feynman (1918–1988),appearing in 1948 as “Space-time Approach to Nonrelativistic Quantum Mechan-ics,” finally exploited the full power of the probability. Feynman took the wavefunction literally in making its phase and amplitude properties of a particle, work-ing in Dirac’s interpretation of quantum states. Using Huygens’ principle fromwave motion, he knew that the change in a wave on interaction is a variation inits amplitude and phase. The new feature is to separate the interval of free motionof a particle from the moment when it interacts with another particle (a collision)or with a field. The motion is reconceived as a propagator, an extension of theconstruction you’ve already seen several times in these most recent chapters: re-member how parallel transport supplied the tools needed for general relativity,invariant transformation groups of linear motions led to special relativity, andthe eikonal was fundamental to Hamiltonian dynamics and the Hamilton–Jacobitheory? The idea is this. Take a wave with an amplitude A and phase φ suchthat:

ψ = Aeiφ.

It doesn’t matter if A is real or complex, the separation of the phase is a convenience(as Hamilton implicitly did). Then the squared modulus, |A|2, is independent ofφ. Now we propagate the amplitudes—not the modulus—as we would for a waveby saying that

[The amplitude at position (x, t)] = [How you get from (x′, t′)→ (x, t)] times [Theamplitude at position (x, t)] summed over all possible paths


or, in symbolic form:

A(x, t ) =∫

(all paths)K (x, t |x′, t ′)A(x′, t ′)D(x′, t ′),

which is now called the path integral. This change in notation is central to theidea, that D is the step along any possible path, not any one in particular. Atthis stage the change doesn’t seem that important since you don’t know whichtrajectory to follow. But, by asserting that the particle “chooses” the path thatminimizes the action, Feynman had the answer. The phase is the action, as itis for Hamilton’s optical eikonal, and since that is quantized, so is the motion.This form of the transport equation was already known from potential theory andhad been used for electromagnetic wave propagation, especially during the WorldWar II in the design of waveguides and antennas for radar. The function K is aGreen function (yes, George Green again!). It can also be written in relativisticform by using the four-momentum the momentum that includes the energy; theformalism is relativistically covariant, which was the crucial step in Feynman’ssearch for a general formulation of quantum electrodynamics. We can work infour-momentum and then transform back, if necessary, to spacetime. If the changein time of the amplitude is an infinite series because (as Zeno would have said)to go from (x′, t ′) to (x, t ) you must pass through an intermediate (x′′, t ′)′, and soon. Feynman realized that the free propagator is actually the first term and thatthe series (which had been studied extensively in the first half of the twentiethcentury in connection with the theory of integral equations and functionals byFredholm and Volterra) is the sum over all possible interactions (paths) that theparticle wave function ψ can follow. This, when expressed by vertices for theinteractions, introduces the potentials that replace forces. The first result was anindependent derivation of the Schrodinger equation. Along with the formalism,which had also been explored by Julian Schwinger, Feynman proposed a graphicaltechnique for representing the integrals, now known as Feynman diagrams. Theseare, essentially, cartoon vector diagrams of the various steps in the propagation.But they are also calculational tools since a set of rules drawn from the formalismtell how the interpret individual parts of the graphs. Each corresponds to a precisemathematical equivalent in the developed perturbation series, the diagrams aremerely pneumonics for the analytical development. But the tool is a remarkablygeneral and valuable one, allowing the visualization of the interactions in a wayNewton could have appreciated. In this way forces become fields only and arecarried impulsively at interactions by particles. For instance, a propagator is asimple line. But there can be interactions and these are drawn as vertices. Thecartoons are just tools, but they enormously simplified the mass of formalism byallowing the potentials and their effects to be visualized.

In this new world of trajectories without forces we still have energy and fields,the two remaining—but altered—classically defined concepts. Instead of havingpoint particles, we trace the paths of the centers of mass as a propagation ofan amplitude. Taking seriously the duality of matter means the wave functionis an amplitude of a probability density. In this case, we can include all of the


mechanical effects without ever dealing with forces. The path of a particle becomesa variable quantity and, using the action and the variational principles it supports,the minimum path within a field can be determined. Then this becomes the “mostprobable” path for a body to follow. There are no forces because we now needonly the potential, a continuous field through which a body moves. For a field, theposition isn’t the important variable but the intensity of the field.

A particle arriving, for instance, at a slit isn’t merely a point mass. The event hasan associated probability of occurring. The slit, analogously to a wave source, actsas a boundary condition on the motion of the particle. Because of the uncertaintyof the particles momentum and location, there is a chance that it emerges fromsomeplace within the slit rather than the center and, as with a wave, we now have themomentum (the direction of propagation) distributed with some probability throughthe space after the slit. This had been experimentally verified by Davisson andGermer in 1927 as a test of de Broglie’s hypothesis. To describe the subsequentmotion we consider the range of all possible trajectories and compute their relativelikelihood, from which we get the most direct path between two points given theconstraint of the energy of the particle and the potential through which it is moved.For a classical particle, as we saw for statistical mechanics, we are dealing witha point mass and the probability is computed by asking what paths the particletakes. For the quantum mechanical case, we ask instead how a wave is propagatedand then sum the amplitude at each point around that central path to find, throughthe quantum conditions, a distribution. In this way, for instance, a double slitproduces a diffraction pattern for the sum of many particles that pass through it.

Feynman showed that the path integral yields the Schrodinger equation for non-relativistic motion and then extended the approach to interaction of relativisticparticles and fields. For the nonrelativistic case it is the probabilistic dynamicsof Brownian motion. Of all paths a particle can take between two points in aspacetime, or a phase space, one is special—the path that minimizes the action.The beginning of the path integral approach was Feynman’s attempt to understanda remark by Dirac that the phase of a wave function is related to the action. In fact,this was already embedded in the mechanics of Hamilton applied to geometricaloptics. Imagine a plane wave. At each point in the wave there’s a phase andthe amplitude propagates in space and time at constant phase (unless the wavedisperses). The same can be thought of a particle in this new quantum world. Inthat case, the past that minimizes the change in the phase is the same one thatobeys the Euler-Lagrange equations, it minimizes the action.

THE MICROSCOPIC FORCES

Quantum mechanics was able to address atomic and molecular structure andthe interaction of radiation with matter using only electromagnetic potentials.But what holds the nucleus together? This was a tremendous difficulty after thediscovery of the “nuclear atom.” Once it was clear that the nucleus is composed ofa bunch of identical positively charged massive particles, protons, their mutuallyrepulsive potential, the Coulomb interaction, must be balanced by something. Thediscovery of the neutron in 1932 by James Chadwick showed the way toward aresolution of the paradox. This neutral particle has almost the same mass as the


proton and nearly always the same number of particles in the nucleus.9 The closematch between the masses led Heisenberg to postulate that the two are actuallystates of a single particle, a nucleon, with the proton being the ground state. Heand Ettore Majorana (1906–1938?) introduced a new sort of angular momentum,called isospin, that behaves similarly to the spin of the electron and proton and, byanalogy, called one particle the “spin up” state and the other “the spin down.” Inthe same way that scalar products of the spin are responsible for a force betweenelectrons, they introduced an interaction that is the scalar product of the isospins.This was the first time a fundamentally new force had been proposed, the strongforce, based on a phenomenological necessity.

But if these two particles are states they should be able to transform into eachother like the spontaneous transitions among atomic states. This was clear fromthe decay of the neutron into a proton. In the process, an electron is emittedthat accounts for both the mass difference and the charge balance. But whenthe sums were done the balance sheet was deficient, the electron—which wasidentified quickly with the β particle of radioactivity—didn’t emerge at a singleenergy. Cloud chamber and ionization studies showed, instead, that it displayed acontinuous rage of energies. An explanation was tentatively proposed by Pauli: thedecay involves the emission of another, neutral particle of low mass and the energyis distributed among the decay products randomly and continuously. Enrico Fermi(1901–1954) jokingly dubbed this “neutrino,” the little neutral thing in contrast tothe neutron (in Italian, “neutrone” or big neutral particle) and in 1934 produceda theoretical explanation for the decay using creation and annihilation operatorsand a pointlike interaction based on an extension of quantum electrodynamics.This was the last new force, the weak interaction that holds individual particlestogether and dominates the electromagnetic forces on the scales even smaller thanthe nucleus. The particle was finally detected directly only in the 1950s but itsbasic properties were already scoped out from the elaborations of the Fermi theoryof β-decay.

The strong force, however, did not seem to be pointlike. Instead, it required arange that was given by the size of the nucleus and the absence of its effects at anygreater distance. An explanation was proposed by H. Yukawa in 1935 using theuncertainty principle. He postulated a virtual particle that is exchanged betweenthe nucleons, independent of their electric charge. By virtual, he meant that theexchange occurs within the uncertainty time based on the mass of the hypotheticalparticle. This required the mesotron or meson to be intermediate mass, nearly thatof the nucleons and far greater than the electron, because of which the strong forcewould look like a pointlike interaction with an exponential cutoff, so he couldwrite V (r ) = [exp(−r/L)]/r where L was the Compton wavelength of the meson.This is the origin of the charge independence of many nuclear properties.

The stage was set for the next unification, finally achieved in the mid-1960sby Steven Weinberg, Abdus Salam, and Sheldon Glashow. They assumed insteadthat the interaction is not pointlike and derived masses for the carrier significantlygreater than that of the proton, about 80 GeV/c2. The weak force is due to twoparticles, Z0 and W (which come sin charged and neutral varieties), each withabout the same mass. A proposal along this line had been made by Klein, onanalogy with the Yukawa construction, to avoid the pointlike interaction required


for the Fermi theory. But the particle mass had to be quite large, much greater thanthe proton’s, because the range is so short and this was proposed in the extensionof the idea along with the charge differences among the proposed carriers of theweak force. The photon becomes the third member of this family, now writtenas (γ , Z0, W ±,0), a step that unifies the electromagnetic and weak forces asthe electroweak interaction. Notice we are still using the word force but we aredescribing the interactions by dynamical potentials. The unification of the baryonsand mesons was achieved by Gell Mann, Ne’eman, and Zweig, who showed that thezoo of massive particles, the baryons, can be reduced to states of a set of only threeelementary particles, called quarks. To this Glashow soon added a fourth. Thesecontrol the strong force through their interaction by a set of spinless particles,called gluons. But to go deeper into the developing picture, which is still far fromcomplete, would take us too far afield—you will have to consult other books inthis series—and here we end our journey.

NOTES

1. Flat screen displays use the polarization effect of electric fields on liquid crystals, asemi-ordered state of matter that polarize and change their optical properties dependingon the local potentials.

2. You can think of a stadium in which the higher levels for the seats require moreenergy to climb the steps and are less populated. The lower levels are filled first and thenumber of seats in each row determines the populations. If there are random exchangesbetween the rows that depend only on a global parameter, the energy, then the rows willbe peopled depending only on the mean energy of the arriving spectators.

3. The story of this paper may be an inspiration to you as a fledgling phycist. Recallingthe event in 1971, Goudsmit recounted: “We had just written a short article in Germanand given to Ehrenfest, who wanted to send it to “Naturwissenschaften.” Now it is beingtold that Uhlenbeck got frightened, went to Ehrenfest and said: “Don’t send it off, becauseit probably is wrong; it is impossible, one cannot have an electron that rotates at such highspeed and has the right moment.” And Ehrenfest replied: “It is too late, I have sent it offalready.” But I do not remember the event; I never had the idea that it was wrong becauseI did not know enough. The one thing I remember is that Ehrenfest said to me: “Well, thatis a nice idea, though it may be wrong. But you don’t yet have a reputation, so you havenothing to lose.” That is the only thing I remember” [from Foundations of Modern EPR,ed. G.R. Eaton, S.S. Eaton and K.M. Salikhov (World Scientific: Singapore, 1998)].

4. The Hamilton–Jacobi equation had been extensively discussed in the Munich circleand included as a central tool within quantum theory in Sommerfeld’s book on AtomicStructure and Spectral Lines. Its several editions made the technique widely known. Withinthe equations, the individual momenta separate. They are represented by gradients of theeikonal and therefore add like phases.

5. Laboratory studies had already demonstrated that the successively lower energyseries of spectral lines have increasing width. This was expressed, e.g., as s, sharp, forthe first (Lyman) series, d diffuse, for the Paschen series. For the Balmer (principal orp series) and the Brackett ( f , fine), the nomenclature is a bit different. These also wereused to designate the orbital states for the Bohr atom.

6. A Hermitian matrix, or operator, is one for which the eigenvalues are real. Thesecorrespond, in quantum mechanics, to the ensemble averages and are the physicallymeasurable quantities.


7. Defined by:

[u, v] =∑

j

(∂u

∂q j

∂v

∂p j− ∂v

∂q j

∂u

∂p j

),

where the sum is over the set of j coordinates appropriate to the phase space with positionsq and momenta p (three dimensional, for instance). Immediately, for arbitrary (u, v) wehave anticommutivity, [u, v] = −[v, u], and linearity, [u1 + u2, v] = [u1, v] + [u2, v].

8. The only odd feature is the discretization required for the J value but that comesfrom the property of the functions that satisfy the Schrodinger equation (the eigenfunc-tions are spherical harmonics and the associated eigenvalues are indexed by integersj ( j + 1)). This and the vibrational quantization actually had already been found from theold quantum theory and it are the only representations that survived the transition to thenew mechanics. From these energies and the statistical weights of the states the popu-lations from collisions—the Boltzmann distribution—can be computed, thus the internalstate of any particles can be specified as a function of temperature.

9. An interesting point is that the nuclei of the chemical elements have the numberof protons, Z—which is the same as the number of electrons and equals the atomicnumber—as neutrons N . An important experimental result is that there is a range allowedfor the difference N − Z that is neither symmetric in neutron or proton richness nor eververy large. In an analogy with thermodynamics, at some point the forces saturate and theexcess particles evaporate from the binding potential.


APPENDIX: SOMEMATHEMATICAL IDEAS

It is interesting thus to follow the intellectual truths of analysis in the phenomenaof nature. This correspondence, of which the system of the world will offer usnumerous examples, makes one of the greatest charms attached to mathematicalspeculations.

—P. S. Laplace, Exposition du Systeme du Monde (1799)

Many of the things you’ve seen in this book require mathematics you may not yethave encountered in your studies. Don’t worry, you will. This appendix isn’t meantto replace either formal study or serve as a textbook on the mathematical methods.Instead, much of modern mathematics has roots in questions related to physics,especially dynamics, which can often be hidden in the formalism of the abstrac-tions. I’m hoping you’ll read this appendix and use it as a sort of essay to set someof these ideas and methods in context. Although this book is about physics, the de-velopment of mathematical techniques in the seventeenth and eighteenth centuryis an essential part of the story since many of its sources lay in physical—oftenmechanical—problems. Nowhere is this more evident than the development of thecalculus.

Motion is either continuous or impulsive, and we have seen the difficultiesthe latter presented to the medievals. They spent considerable ink on whether itwas possible, or reasonable, to divide any interval into infinitely many smallerones. For Zeno, the pre-Socratic philosopher whose paradoxes of motion wereactually considerations of the infinite and infinitesimal, it was impossible toimagine the subdivided interval because an infinite number of intervals wouldrequire an infinite time to cross them all. Archimedes was able to addressthe problem with a first pass at the concept of a limit, and Cavalieri, Galileo,Newton, and Leibniz were finally able to exploit it to describe extension andmotion.

204 Appendix

FUNCTIONS AND SPACES

For the purpose of mechanics, we can take a trajectory as an example of a func-tion. Speaking classically, in the language Newton, Lagrange, Laplace, and theircontemporaries found familiar, t is an ordered set of coordinates in time. If aforce is either smoothly varying or constant, the curve is at least continuous inits second-order changes, those that compare where the particle was at some timet − 2�t with now t . The velocity is also continuous to the same order becauseto go from x(t − 2�t ) to x(t ) you need to know the velocity at the two previoustimes that take you from x(t − 2�t ) to x(t − �t ) and then to x(t ). Making thismore precise required almost a century’s investigation, culminating with Cauchy’sdefinition of continuity. Yes, you can think of this as a relation between two vari-ables, one independent and the other dependent, or as a track on a graph. Withinmechanics the change from continuous to discontinuous motion, the importanceof collisions, forced a generalization of the concept of function beyond this origi-nal form of continuity. This is connected with the way change, differentiation, isdefined as we’ll discuss. But if you’ve already seen calculus and know that thetangent to a line is the derivative of the function represented by the graph, you cansee one way this can happen. In a collision, the change in speed and direction fora particle is seemingly instantaneous. It’s true that for real-world cases such asautomobile collisions, this isn’t strictly true. But it is close enough, for a physicist,because we can make the interval during the collision as short as we’d like andexamine only the state going in compared to the state going out of the site of theaccident.

THE CALCULUS

Think of a small piece of a path, imagining it in one dimension, with a lengthL. If this is crossed in a time interval T then the speed is, of course, L/T ,right? Not precisely. As you saw from the Merton rule, the speed may vary duringT and the time interval can itself become very short. You can imagine, then,an interval �t in which the speed is constant, but in that case the distancetraversed will be �x = V �t only during the interval. If we imagine V changing,then at a time t + �t it will be V (t + �t ) while it was V (t ) at the start andtherefore the acceleration, the change of the speed in a time interval �t , isa = [V (T + �t ) − V (t )]/�t = �V /�t if you take the intervals to be smallenough. For a constant acceleration, or a constant speed, we can take the intervalof time to be as small as you want, as long as the difference in the position andthe velocity decrease at the same rate as �t decreases. This is the basic idea of aderivative or, as Newton would call it, the fluxion. If the speed, or the space, changescontinuously (and that notion is a tricky one that would exercise mathematiciansfor nearly two centuries), we have a device for calculating the rate of change. Fora curve in, for example, two dimensions, (x, y), the curve is y(x) and the rate ofchange of one coordinate relative to another is �y/�x. The continuity is assuredwhen we can take the limit as �x → 0 and have the interval in y change withoutbecoming infinite because �y → 0 at the same rate. Newton used a dot to denote

Appendix 205

this quantity when it referred to space and time; a more general way of writingthis is:

V (t ) = lim�t→0

�x

�t≡ dx(t )

dt≡ x(t )

and similarly for acceleration. This does not mean that the speed, or acceleration,is necessarily constant everywhere and forever, merely that it remains constantover an infinitesimal interval, which I hope seems intuitive.

If we imagine that a quantity depends on many things, for instance threedifferent directions in space as well as time, we can ask how it changes when onlyone of these is changed at a time. For instance, the area of an object dependson two coordinates, the volume on three, etc. So we can imagine a generalizedsomething, call it now a function F (x, y , z), for instance, and then the change ofF with respect to only one of the quantities on which it depends, say x, is

lim�x→0

F (x + �x, y , z) − F (x, y , z)

�x≡ ∂ F

∂x

the other two remaining fixed. This is the partial derivative and it is going to be anessential feature of all subsequent discussions of how fields change. Along withthe changes, we have the cumulant, the path itself. This is a different problem, tofind a way to sum the tiny intervals covered in each interval of time to produce afinite distance. This is the integral, first introduced by Leibniz,

L = limN→∞

N∑k=0

V (tk)�tk ≡∫ T

0V (t )dt .

The sum is then the result of infinitely many infinitesimal intervals during which thespeed remains constant, but allowing that the speed may be continually changingfrom one tiny interval to the next. It is a modern notion that the integral isthe inverse operation to derivation, but you can see how it happens. The term“quadrature” refers to taking the integral, while for a geometric discussion whenlength and not time are involved, the derivative is the tangent to the curve. Theidea of a variation is closely related to a differential, the unspecified change of acontinuous quantity δt , for instance, that can be made as small as we’d like but isotherwise unconstrained. This was already implicit in statics, from Archimedes andLeonardo. Remember that the lever requires the concept of a virtual displacement,where we imagine taking weights and lifting it at one end while lowering it at theother by some angle δθ . What we’re now doing is making this a variable

Motion of individual particles can be extended to the behavior of continuousmedia with the extension of force and so the mathematical machinery also needsto be modified. Now, instead of a trajectory—a line—in space marking the motionin time, we can have simultaneous variations of many points on a surface or avolume at the same time, all parts of which are connected to each other. This is theorigin of partial differential equations. The variation of a distributed quantity, say

206 Appendix

something that depends on two coordinates (x, y), we can take the change in eachdirection separately keeping the alternative position fixed (that is, the change ofthe function at one position with respect to the other):

� f (x, y) =[� f

�x

]y

�x +[� f

�y

]x

�y →(

∂ f

∂x

)y

�x +(

∂ f

∂y

)x

�y.

This can be extended to higher order derivatives as well distributed in space.The close link between differential equations and dynamics had already beenrealized by Euler and Lagrange in the eighteenth century, but was exploited farmore in the next. The connection is clear from the second law of motion, thatthe rate of change of a velocity with time—the acceleration, that is the secondderivative of the displacement—is produced by a force that can change in space.It was, therefore, a short step to associate any second-order differential equationwith forces.1 The important idea is that a line, surface, or volume can changecontinually but not identically in both space and time. It depends on the initialand boundary conditions. Much of the theory related to these equations in thenineteenth century involved studying which conditions lead to which types ofsolutions, whether these are uniquely specified by the boundary conditions, andhow they evolve. Fourier was the first to show that the solutions to these equationscan be expressed as a series of superimposed waves, periodic functions in theindependent coordinates. For example, for a string this is an infinite series of sinesand cosines that are the harmonic functions of modes of the string (the harmonicsof the fundamental frequency). He showed how to use the boundary conditions toobtain the amplitudes—thus determining the phases—of the waves. The methodis called the Fourier integral or, more frequently, the Fourier transform. It was acomplementary representation to one used by Laplace to solve field problems forgravity, the Laplace transform.

VECTORS AND TENSORS

As we discussed with medieval kinematics, the problem of describing how a bodymoves was especially vexing when it required more than one dimension. Thedifference between speed and velocity is that the latter involves a direction. Thequantity is called a vector, a term coined by William Rowan Hamilton in themid-1800s long after the thing itself was first employed. But how can you quantifythis “object”? The components of a force, or a velocity, are, as Newton showedin the Principia, combined using a parallelogram. The resultant of two orthogonalcomponents lies along the diagonal of the resulting figure and the Pythagoreantheorem gives its magnitude. Inversely, for any coordinate system to which thecomponents are referred, the single vector is decomposable into its componentseach of which points along some axis that is represented by a unit vector, n forinstance for some normal vector, that has a magnitude of unity and signifies thedirection. For example, a wind speed may be V km hr−1 from the northeast, butthis can also be described as a vector V = V N N + V E E where V E and V N arethe components. This is a different sort of addition. The two directions, north andeast, remain distinct. Hamilton’s 1842 realization was the analogy with complex

Appendix 207

numbers, for which the real and imaginary components remain distinct and thatsum quadratically in magnitude, and how to also multiply these quantities. Thereare two separate forms for this. One is the projection of one vector onto another,the result then being only a magnitude and thus called a scalar. Represented fortwo vectors A and B by C = A · B it doesn’t matter which vector is projected ontowhich. On the other hand, recall our many discussions of the lever. A beam liesalong some direction and a force applied perpendicular to its length produces arotation about an axis that is perpendicular to the two of them. The torque, theforce that produces the angular rotation, is given by the vector product, T = r × Fwhere F is the force on the beam; the angular momentum is r × v for a transversevelocity v.2 An important, indeed fundamental property of this product is thatit doesn’t commute. If you apply the produce the other way round it reversessign (the rotation goes in the opposite sense). This decomposition also applies tochanges. A gradient is the directional change in a quantity and is, therefore, avector. But a vector quantity can also change in each direction and this requires afurther generalization to a tensor, another Hamiltonian. More general than vectors,since these describe planes as well as lines in space (for instance, you can say“in a direction” or “lying in the plane”), the foundations were laid somewhatlater in the 1890s through the first decade of the twentieth century by GregorioRicci-Curbastro and Tullio Levi-Civita under the name “absolute differentialcalculus.” An earlier form, the dyadic, was introduced by Josiah Willard Gibbsto describe strains, stresses, and fields. A simple way of imagining the thingis to mark a bunch of points on a sheet of plastic wrap and then stretch thesurface along some arbitrary direction. The points displace differentially in thetwo directions (or three if the surface buckles). You can see how this would applyto the generalization of force from Newtonian to relativistic gravitation, and alsohow this generalizes the application of a force to a deformable body. A tensor hasmultiple components as does a vector but these can mix depending on their senseof displacement. So something pointing in the NE direction can have componentsthat both change along the N and E directions, for example. So, for instance, atensor T can be written with its components along, say, two directions simulta-neously,

T = T N N NN + T N ENE + T E N EN + T E E EE + · · · .

The individual components do not have to be symmetric (e.g., T E N doesn’t nec-essarily have to be the same as T N E . Think of a vortex shearing that has a senseof helicity and you can see why this might be so).

VARIATIONAL CALCULUS

For Fermat, passing light through a refracting medium was a minimization problem,the problem was to find the path that minimized the transit time or one of minimumaction of the medium. If the speed of light in a vacuum is a constant but the speedwithin anything else is lower, a continuously changing index of refraction is thesame as a continuously varying speed of propagation. Then the path of minimumlength may be other than a straight line (which would be the result for a uniform

208 Appendix

medium). As an interesting combination of differential and integral methods, thisquestion can be addressed without any recourse to dynamics. The length of anysegment of the line is d s, say, but the total length is S = ∫

d s between any twofixed boundary points. The path can be two- or three-dimensional. The minimumlength is the same as the minimum time and this is the connection with dynamics.If we ask, instead, what will be the minimum time to move between two pointsunder the action of a force, we have the brachistochrone problem, which wasposed as a challenge to the continental mathematicians by Newton and the RoyalSociety soon after the appearance of the Principia. The problem is essentially thesame, if the force is a function of space, to find the path corresponding to thetotal acceleration experienced by a particle. Then, for a statics problem, the twoapproaches can be combined to find the curve of minimum length hanging underthe action of a constant external force (the catenary curve).

In the second half of the eighteenth century, this problem was generalizedfollowing Leibniz’s ideas of vis viva and action. Lagrange realized that the principleof least action could be geometricized along the same lines as the brachistochrone,to find the path that minimizes the action required for motion between two fixedpoints in space and/or time. I’ll go into a bit more detail here just to show youhow such a calculation looks and how the formalism expressed the physical ideas.Taking a (potential) field that doesn’t explicitly depend on time and is independentof the velocity of the particle, the force being the gradient of the potential, theacceleration depends on the change of the momentum in time and is given by achange in the potential in space. He realized he could write the generalized formfor the momentum:

p = ∂T

∂ x,

which is read as “the momentum along some direction, say x, is the changein kinetic energy (vis viva) with respect to the velocity in the x direction,” andcombine this with the equation for the force:

d

dt

(∂T

∂ x

)= −∂V

∂x,

which reads “the force is the rate of spatial change of the potential (field) alongthe same direction,” to obtain a new, universal function:

L = T − V

now called the Langrangian function, from which the equations of motion couldbe rewritten in terms of this single function:

d

dt

(∂L

∂ x

)= ∂L

∂x

Appendix 209

for each component of the system of coordinates. But more important was hisrealization that this equation is a result of the same minimization principle thattakes the action and asks what trajectory produces a minimum value for

S =∫ t1

t0

Ldt

during an interval of time �t = t1 − t0, that is,

δS = 0

expresses the least action principle in compact form. The energy need not beconstant for this path, and because the function is a scalar it is independent of thecoordinate system (its form changes but the two components, potential and kineticenergies, remain distinct and it permits interactions between the particles). Thishas become the cornerstone of modern physics. Notice the difference between themomentum and velocity, the latter is the “primitive” variable while the formeris derived from the derivatives, which generalizes the concept of the particlemomentum by allowing even massless particles (e.g., light in the sense of photons)to have a momentum and, in the relativistic framework, allows us to work in anycoordinate frame. Of course, we must confirm that this extremum, δS , is really theminimum action, which can only be verified by showing that the second variationis negative, δ2S < 0. Weirstrauss and the Berlin school of mathematical analystsmade this a central feature of their studies by generalizing the Lagrangian tofunctions in general, founding a whole branch of mathematics called variationalmethods that is no longer directly connected with the dynamical origins of theproblem.3

The major change in perspective was Hamilton’s introduction of a canonicalrepresentation for the momentum and position using the energy (the Hamiltonianfunction) instead of the action as the fundamental link. Although using the leastaction principle, Lagrange had introduced the idea of a derivative function as themomentum, Hamilton fully exploited its possibilities by associating two ordinarydifferential equations (in vector notation):

dpdt

= −∂ H

∂q

for the momentum evolution and

dqdt

= ∂ H

∂p

for the generalized velocity. For Hamilton the position and momentum, not thevelocity, were the primitive variables (unlike Lagrange and Euler). Notice I’vecalled the first the momentum “evolution,” not the force. This is the importantchange. For any particle, each point in space has an associated momentum andboth are needed to specify a trajectory (ultimately q(t )). Then these can become

210 Appendix

the characteristics, analogous to those Monge had introduced to describe surfaces,along which the particle moves. If the total energy is conserved, the set of coordi-nates describes fully its history. Actually this is not that different from Lagrange’soriginal conception of the function that bears his name except the Lagrangianis not a conserved quantity while, if H is independent of time, the Hamiltonianfunction is. Since these together, for a closed path (an orbit), require only thephase instead of the time, the space they collectively describe is called a phasespace and this was taken over directly into statistical mechanics by Boltzmann andGibbs.

GROUPS

Although a sidelight in our development, the nineteenth century also witnessedan explosive growth in mathematical physics, unprecedented in the previoustwo centuries. New viewpoints emerged, linking a broad range of mathematicalideas. One of the most fruitful for the concept of force, and dynamics, was thestudy of transformation groups that formed the nucleus of Sophus Lie’s (1842–1899) contributions to mathematics. We can develop this rather simply from theHamiltonian perspective. In that approach, a particle’s trajectory is the evolutionof its momentum and position through time. These are linear differential equationsthat depend only on the potential.

Lie’s theory begins with the linear transformation, one that takes the value ofa function at some “place,” x, and asks what the value is at x + δ. This is justthe definition of the first derivative, the rate of change of the function is d f /dx.But we can look at this as well in terms of an operation, one that takes f (x) andreturns f (x + δ). For a very small displacement in the dependent variable, thisis developed only to first order in which case, if we say there is a transformationTδ then Tδ f (x) = f (x + δ) and some second Tδ′ that is applied to this secondfunction takes it to δ′ so that Tδ+δ′ = Tδ′ Tδ. The inverse of this transformationalso exists, the path is reversible, so there is also an operator T −1

δ for every Tδ.The product of these two is the identity operator, the function is left unchanged.Now if we change our language and think in terms of motion, you’ll see this isanother way to look at the dynamics. It’s easy to see that if we’ve chosen a line,the transformations are also commutative, it doesn’t matter in which order we takethem and the same is true for their inverses, and any other combination. For asurface, however, we don’t necessarily get the same result. It depends what wemean by a transformation. If we’ve moved the particle on a two-dimensional path,the velocity vector is always tangent to the curve. Let the surface be a plane and thepaths be orthogonal displacements on two arbitrary paths. Although we may arriveat the same point in the end, the velocity is not necessarily in the same direction.Now if we add the restriction the curve is closed; displacing around a closed curve(an orbit) on this plane leaves the vector pointing in the same direction as at thestart. Now let’s do this on a curved surface. Then we get a rotation of the vector,as Lie’s idea was to replace the specific derivative operator with one that is moregeneral, a “displacement.” His treatment followed on the study by Plucker in the1860s and Riemann at around the same time, realizing that translations (in spaceand time) have an interesting mathematical structure.

Appendix 211

Dynamical symmetry arguments were also used by Einstein in his 1905 intro-duction of relativity, what has since become known as the Lorentz group. Thisset of transformations in spacetime satisfies the group properties of invertability,association, and also commutivity. It sits at the core of the theory, the constantspeed of light providing the means for going between reference frames. If insteadof writing the operation out explicitly we use � to represent the transformationand γ to be the governing parameter (this is the factor γ = 1/

√(1 − v2/c2) in

the Lorentz transformations), we have the property that for two frames in relativemotion u = (v1 + v2)/(1 + v1v2/c2) for the velocity addition law and for succes-sive transformations � = �2�1 if we go from one frame to another. In anotherway, simpler static, geometric symmetries are obvious in crystals and during theearly modern period these were likened to the Platonic solids, thus to ideal forms.Kepler used this in his Mysterium Cosmologica to imbed the orbits of the plan-ets in the five solids and wrote treatises on the symmetry of snowflakes and thedodecahedron. This symmetry of crystals is a basic feature of quantum mechan-ics applied to the solid state. It is based on a theorem due to Felix Bloch, thata free electron is sort of confused when being propagated in a periodic struc-ture and can be expanded in an elementary set of periodic functions on thescale of the lattice. The same is used for computing the vibrational spectrum ofcrystals, railroad bridges, and other composite structures that possess specificsymmetries.

DIFFERENTIAL GEOMETRY: ILLUSTRATEDWITH THERMODYNAMICS

Producing maps was a preoccupation, indeed a mania, for the imperialist ex-pansionists of the eighteenth and nineteenth century. Surveying was a highlydeveloped technical art. But the maps were another matter. Coordination of mea-surements on a complex terrain is hardly as simple a problem as measuring thealtitude and bearing of a set of points on the surface. In particular, projective ge-ometry developed to respond to this immediate problem and then was generalizedwhen the mathematical problem appeared far more general than the specific appli-cations. For instance, on a sphere, how do we take two regions measured on whatlooks like a flat plane and then distort them appropriately and match their bound-aries to form a continuous surface? As you saw in the description of relativity,this later, through the ideas of parallel transport and curvature, became funda-mental tools for studying trajectories and led to the tensor formalism of generalrelativity.

We can use a remarkable feature of thermodynamics, its connection with ge-ometry in an unexpected way, to illustrate how differential geometry is appliedto a physical problem. Small thermal changes, calorimetry, were described in themid-nineteenth century using linear equations and this could easily be adapted tothe statement of the first law, that a quantity of heat added to a system producesboth changes in the internal state of the system and work. Now since the internalenergy depends only on the temperature, a cyclic change should leave it at thesame value, just as a path in a gravitational field brings us back to the same place.For that reason it is called a potential. In contrast, the work depends on two state

212 Appendix

variables, pressure and volume, and around a closed path the total work is due toa change in the heat content, not just the internal energy. These can be thought ofas coordinates, as in an indicator diagram (see the discussion of Carnot’s cycle),a reference system in which surfaces represent free energies and entropy. Smallchanges are separable, and as long as the processes are reversible we can use thecontinuity conditions available for any continuous surface, exactly as in geometry.This geometricization, surprisingly, furnishes a closer link to dynamics than theusual conceptions and was employed by both Gibbs and Maxwell when describingenergies. Gibbs actually sent Maxwell a model surface for the thermal properties ofwater, which shows not only the complex shape of the functions needed to describethe processes but also the trajectories followed by specific processes. When wedescribed a change in heat, �Q, as something that produces work, we can recallthat if P is the pressure, V the volume, T the temperature, and U the internalenergy,

�Q = dU − dW = AdT + BdV = dU (T ) + P dV

so the independent coordinates are (T , V ) and the length of the infinitesimaldisplacements become the unit vectors. You’ll notice the close correspondencewith Lie’s ideas of operators. Continuity says when we look at the rates of changerelative to the two coordinates, it doesn’t matter in which order the changes aretaken. Thus, in the language we described for partial derivatives,

[∂ A

∂V

]T

=[

∂ B

∂T

]V

where the subscript tells which quantity is kept constant during the change. Nowfrom the second law of thermodynamics we have a new way of writing the heat,�Q = T d S where now S is the entropy, so a surface of constant entropy is one forwhich we move along a trajectory driven only by T and V since the pressure, P , isa function of only these two. You’ll notice this has the same features as the previousdiscussion of vectors and tensors. Green’s insights on potentials again allow usto see that d S is the normal to the entropy “surface” and adiabatic surfaces arethe same as equipotentials. We can then form a variety of such thermodynamicquantities, the free energy, F = U + P V − T S , the enthalpy, H = U + P V ,and the Gibbs free energy, G = F − µN , using the variable µ to represent thechemical potential (the energy required to change the number of particles in thesystem). In each case, we have a different set of state variables and can transformbetween them precisely as we would between coordinates in different referencesystems. The way we’ve written these should look now like vectors and tensors.I chose this example precisely because it’s so comparatively distant from yourusual notions of changes in space and time and to show the conceptual basisof any field theory. But these are not merely formal analogies. Any change canbe visualized within a space, this was emphasized by Poincare at the end of thenineteenth century. The modern terminology for chemical reactions is landscape,which use both the quantum mechanical energies and the thermodynamic conceptsto describe the favorable states. The rest of the connection between dynamics and

Appendix 213

differential geometry comes from saying that a thermal system follows a path of leastaction and the equilibrium states are given by the minima on the thermodynamicpotential surfaces. Such a construction was attempted by C. Catheodory in the1920s and led to an axiomatization of the mathematical structure.

NOTES

1. This would later prove a powerful analytic tool guiding the construction of analogcomputers that, by judicious couplings of pulleys and springs, later gears, and finallyvacuum tubes, would allow the solution of specific individual and systems of equations.In this way, the concept of force was generalized to include a purely analogical role thatnevertheless proved essential.

2. The modern notation was introduced by Gibbs at the end of the nineteenth centurywhen vector analysis replaced quaternions. This is also written as a “wedge product,”τ = r ∧ L, depending on the cultural habits of the authors. The notation was introducedby Hermann Grassmann at around the same time as Hamilton in his analysis of forms.

3. As a more detailed illustration, for those of you who have seen some calculus,I’ll treat a simple but fundamental problem: how to use the variational method to solvea lever or Atwood machine problem. The principle of least action, and the extensionof the Atwood machine, unites angular momentum and the lever in a way that extendsArchimedes’ reasoning and generalizes the result. We consider two masses m1 and m2

on opposite sides of the fulcrum of a lever at distances r1 and r2. The most importantfeature of this simple picture is that the two masses are constrained to move, as in theAtwood machine, along some path governed by a common property. In the latter case, thelength of the cord connecting the masses doesn’t change and this is the same exampleused by both D’Alembert and Lagrange in the block and tackle couplings with whichthey illustrated virtual displacement. For a level it’s more obvious. The two masses movethrough a common angle, φ, provided the lever remains rigid. The vertical distancesmoved are, however, different because their distances from the pivot point are not equal.For a simple Archimedian example, the masses are not dynamical and the equilibrium isobtained by forcing the equality

m1r1g = m2r2g ↔ W1r1 = W2r2.

The transformation from force to potential energy makes clearer the Greek mechanician’sfundamental axiom. Note that this is true for an arbitrary displacement no matter howsmall. The extension comes when the least action principle is applied to the dynamicalsystem. The kinetic energy is quadratic in the angular displacement so

T = 1

2

[m1r 2

1 + m2r 22

](φ)2

where φ = ω, the angular velocity (or frequency), again for any arbitrary angular displace-ment and

P = (−m2r1 + m2r2)gφ

for the potential energy. The signs of the two masses, although they must be opposite,are arbitrary. Notice that imposing the constraint is that the angular displacements are thesame, that is we have a holonomic system; we arrive at

(m1r 2

2 + m2r 22

)φ ≡ I α = −(W1r1 + W2r2)φ

214 Appendix

which is the equation of motion for a barbell with an angular acceleration α. It was thensimple to extend this to any arbitrary number of masses extending along the rod with thesubsequent definition of the moment of inertia, I , as

I =∑

i

mi r2r →

∫r 2dm,

where the integral is the continuous limit of the mass distribution. This also yields theradius of gyration since the system is rigid and this can be defined independent of time.The formalism thus connects immediately to the motion of a pendulum for which thelength is only r1 and with only a single mass and yields a harmonic oscillator with theusual well-known frequency. Although hardly still necessary at the end of the eighteenthcentury, this also clearly resolved the question of the equilibrium of, and oscillation of, abent beam since it only requires a change in the angles for the calculation of the momentof inertia.

TIMELINE

5th–4th century BC Plato and the school of Athens; Eudoxus of Cnidus, firstgeometric-kinematic model of planetary motion; EuclidElements, compilation of hellenistic geometry; Aristo-tle (Physica, de Caelo), laws of motion and nature ofspace and time; Democritus and Epicurus of Samos andatomism

3rd century BC Archimedes, hydrostatics, mechanical principles of equi-librium; Apollonius of Perga, kinematical theory ofepicyclic motion for planetary astronomy; Aristarchusof Samos, first suggestion of a heliocentric arrangementof the planets

2nd century BC Hipparchus and precession1st century BC Lucretius, de Rerum Natura, atomistic philosophy1st century AD Vitruvius; Hero of Alexandria2nd century AD Claudius Ptolemy, Almagest, compilation of hellenistic

mathematical planetary astronomy800–900 Thabit ibn Qurra’s Kitab fi’l-qarastun (The book on the

beam balance)1000–1037 ibn Sina (Avicenna)1100–1200 ibn Rushd (Averroes); John of Seville translates Almagest;

Gerard of Cremona active in translation; Maimonedes1200–1237 Jordanus de Nemore, positional gravity and the origin of

medieval statics1210, 1277 Two principal condemnations of non-Aristotelian philos-

ophy issued at Paris1268–1286 William or Moerbeke’s translations of Greek works on

mathematics and mechanics

216 Timeline

1320–1450 Thomas Bradwardine and the Merton school of “calcu-lators,” laws of motion, the mean value theorem (“theMerton Rule”) for displacement and speed in time; JeanBuridan, Nichole Oresme and the Parisian school of me-chanics, introduction of graphical methods, studies of thelaws for continuous and accelerated motions; Nicholas ofCusa, problem of relative motion of the observer and mo-tion of the earth.

1470–1505 Leonardo da Vinci and applied mechanics1472 von Peuerbach’s Theoriae Novae Planetarum.1492 Columbus’ first voyage to the Americas1542 Copernicus’ de Revolutionibus Orbium Coelestium, intro-

duction of a consistent heliocentric planetary model1564 birth of Galileo1588 Tycho’s De mundi aetherei recentioribus phaenomenis,

publication on the comet of 1577 and the introductionof a composite helio/geocentric cosmology

1600 Gilbert’s de Magnete1609 Kepler’s Astronomia Nova1610 Galileo’s Sidereus Nuncius1618–1621 Publication of Kepler’s Epitome astronomiae Coperni-

canae; Galileo’s il Saggiatore1632 Galileo’s Dialogo sopra i due massimi sistemi del mondo1638 Galileo’s Discorsi e dimostrazioni matematiche intorno a

due nuove scienze1642 Death of Galileo, birth of Newton1644 Descartes’ Principia Philosophiae; Torricelli’s announce-

ment of the invention of the barometer1647 Pascal’s demonstration of atmospheric pressure gradient1658 Christian Huygens’ description of the pendulum clock

and centrifugal force1662 Boyle’s law for gas pressure and studies of vacuum and

gases1665–1666 Newton discovers the inverse square law for gravitation

(the annus mirabilis)1672 von Guericke’s demonstration at Magdeburg of atmo-

spheric pressure1676–1689 Leibniz’ dynamical investigations, especially principle of

vis viva1679 Hooke’s law presented in the Cutlerian Lectures1687 Newton’s Principia Mathematicae Naturalis; 2nd edition

1713; 3rd edition 1726; first English translation (AndrewMotte) 1729

1695–1697 Halley’s papers on the tides1704 First edition of Newton’s Optiks1712 Newcomen invents the modern steam engine1725 Discovery of abberation of starlight by Bradley

Timeline 217

1727 Death of Newton1734 Voltaire’s English Letters (Lettres philosophiques) that

popularized the Newtonian philosophy on the continent1736–1737 Euler’s Mechanica1738 Publication of Daniel Bernoulli’s Hydrodynamica,

derivation of Boyle’s law, and first comprehensive treatiseon fluid dynamics

1743 D’Alembert’s Traite de dynamique and the principle ofvirtual work

1747–1760 Franklin’s experiments and studies on electricity1748 Maclaurin’s Account of Sir Issac Newton’s Philosophical

Discoveries1750–1780 Debate on the nature of charge1754–1756 Lagrange and Euler’s foundation of calculus of variations1758 Return of the comet of 1682, now called “Halley’s comet’;

publication of Boscovich’s Theoria Philosophae Natu-ralis’

1759 Aepinus publishes the two-charge theory of electricity1769 Watt improves the Newcomen design with a separate

condenser1775 Maskylene’s measurements of the density of mountains,

foundations of geodesy1781 William Herschel discovers Uranus1783 Death of Euler1784 Coulomb torsion balance experiment on electric charge;

Atwood’s description of the simplest mechanical device1788 Lagrange’s Traite de mecanique analytique1789 Lavoisier’s Elements of Chemistry1798 Cavendish experiment, determination of the density of

the Earth (and the Newtonian gravitational constant)1799 Beginning of the publication of Laplace’s Mecanique

Celeste1813 Gauss’ law for fields; publication by Poisson of the field

equation for a matter/charge distribution1820 Oersted’s discovery of electromagnetism1821 Publication by Biot and Savart of the law for electromag-

netism1821–1823 Amprere’s publication of the laws of electrodynamics1824 S. Carnot’s On the Motive Power of Fire1827 Green’s publication of Essay on Electricity and Mag-

netism; Ohm’s explanation for circuits and currents1828 Brown’s paper on microscopic motion of pollen (Brownian

motion)1831 Faraday and Henry discover electromagnetic induction

and self-induction1831–1851 Faraday’s Experimental Researches in Electricity and

Magnetism

218 Timeline

1834–1835 Hamilton’s On a General Method in Dynamics1842 Hamilton’s discovery of quaternions and vector analysis1843 Joule’s determination of the mechanical equivalent of

heat1846 Discovery of Neptune, predicted by Adams and Leverrier

based on Newtonian gravitational theory1847 Publication of Helmholtz’s The Conservation of Forces1851 Foucault’s demonstration of the Coriolis effect and rota-

tion of the Earth1852 Invention of gyroscope; Faraday publishes the idea of

“lines of force”1857 Maxwell publishes On Faraday’s Lines of Force; publica-

tion of Clausius’ A Kind of Motion We Call Heat; Kelvinand Helmholtz determine the lower limit to the age of theSun

1860 Maxwell’s On the Dynamical Theory of Gases1862 Maxwell’s On the Dynamical Theory of the Electromag-

netic Field1867 Thompson and Tait’s Principles of Natural Philosophy1871–1906 Boltzmann’s work on statistical mechanics, Lectures on

Gas Theory1872 Maxwell’s Treatise on Electricity and Magnetism1881–1887 Michelson, along with Morley, determines invariance of

the speed of light1883–1901 Mach’s Principles of Mechanics (four editions)1884 Poynting’s paper on radiation pressure1887–1890 Poincare on three-body problem and orbital stability,

chaos in dynamical systems1892 Lorentz’s paper on the electrodynamics of moving bodies1898 J. J. Thomson’s determination of the mass and charge of

the electron1900 Planck introduces quantized statistics for blackbody ra-

diation1902 Gibbs publishes Elementary Principles of Statistical Me-

chanics1905 Einstein’s papers on Brownian motion, special relativity

theory, and first paper on the photoelectric effect1907 Minkowski introduces spacetime, the unification of space

and time1909 Perrin measures molecular trajectories using Einstein’s

theory of Brownian motion; Milliken and Fletcher’s oildrop experiment for the elementary unit of charge

1911 Rutherford’s scattering experiments; Einstein derives theprecession rate of Mercury’s orbit

1913 Bohr’s papers on atomic structure and the Correspon-dence Principle, On the Constitution of Atoms and

Timeline 219

Molecules; Einstein and Grossmann’s Entwurf paper ongeneralized gravitation theory

1916 Einstein publishes the final General Theory of Relativ-ity; Schwarzschild obtains first solution for a nonrotatingpoint mass

1919 Measurement of deflection of light during a solar eclipse1919–1923 Sommerfeld’s Atomic Structure and Spectral Lines1922 Compton’s discovery of dynamical X-ray scattering,

Compton effect1923-1924 de Broglie’s thesis on wave-particle duality1925 Heisenberg’s paper on quantum mechanics1925–1926 Born, Heisenberg, and Jordan’s papers on quantum me-

chanics; Heisenberg’s uncertainty principle1926–1927 Schrodinger’s foundational papers on wave mechanics1927 Bohr introduces complementarity1928 Dirac’s equation for relativistic electrons and the begin-

ning of quantum electrodynamics1930 Dirac’s Principles of Quantum Mechanics (3rd edition

1965); Born and Jordan’s Elementare Quantenmechanik1932 Discovery of the neutron1934 Fermi’s theory of β-decay and the weak interaction1934–1970 Ingredients of the “Standard Model” for electroweak uni-

fication1935 Yukawa’s meson theory of nuclear structure1948 Feynman’s paper introducing the path integral formalism

for quantum mechanics1974 Discovery of the binary pulsar 1915+21 and gravitational

wave emission1979 Discovery of cosmological scale gravitational lenses


BIBLIOGRAPHY

Agazzi, J. 1971. Faraday as a Natural Philosopher (Chicago).Aiton, E. J. 1972. The Vortex Theory of Planetary Motions (New York: Neale

Watson).. 1987. “Peurbach’s Theoricae Novae Planetarum: A Translation with

Commentary.” Osiris, 3, 4.Aristotle. 1930. Physica, de Caelo (trans. W. D. Ross) (Oxford: Oxford University

Press).. 1936, Minor Works: Mechanics (trans. W. S. Hett) (Cambridge, MA:

Harvard University Press)Atwood, G. 1794. “Investigations, Founded on the Theory of Motion, for Deter-

mining the Times of Vibrations of Watch Balances.” Phil Trans R. Soc., 84,119.

Blackwell, R. J. (ed.) 1986. Christian Huygens’ “The Penulum Clock or GeometricalDemonstration Concerning the Motion of Pendula as Applied to Clocks” (Ames:Iowa State University Press).

Boltzmann, L. 1995. Lectures on Gas Theory (ed. S. G. Brush) (New York: Dover).Boscovich, S. J. and Roger, J. 1763. A Theory of Natural Philosophy (trans. J. M.

Childs) (Chicago: Open Court) (ed. 1922).Brush, S. G. 1976. The Kind of Motion We Call Heat: A History of the Kinetic

Theory of Gases in the 19th Century (2 vols.) (New York: American Elsevier).Brush, S. G. and Hall, N. S. 2003. The Kinetic Theory of Gases: An Anthology of

Classic Papers with Historical Commentary (London: Imperial College Press).Carmody, T. and Kobus, H. (eds.) 1968. Hydrodynamics, by Daniel Bernoulli, and

Hydraulics, by Johann Bernoulli (New York: Dover).Casper, M. 1959. Johannes Kepler (New York: Dover).Chandrasekhar, S. 1967. Ellipsoidal Figures in Equilibrium (New York: Dover).

. 1995. Newton for the Common Reader (Cambridge, UK: Cambridge Uni-versity Press).

222 Bibliography

Clagett, M. 1959. The Science of Mechanics in the Middle Ages (Madison: Universityof Wisconsin Press).

. 1964. Archimedes in the Middle Ages (multiple volumes) (Wisconsin:University of Wisconsin Press) (Philadelphia, PA: American PhilosophicalSociety).

Cohen, I. B. 1980. The Newtonian Revolution (Cambridge, MA: Harvard UniversityPress)

. 1985. Revolutions in Science (Cambridge, MA: Harvard University Press).

. 1991. The Birth of the New Physics (2nd ed.) (New York: W. W. Norton).Cohen, I. B. and Drabkin, I. E. 1948. A Source Book in Greek Science (Cambridge,

MA: Harvard University Press).Cohen, I. B. and Whitman, A. (eds.) 1999. Isaac Newton—The Principia: Math-

ematical Principles of Natural Philosophy (Berkeley: University of CaliforniaPress) (also see Cohen, I. B. and Koyre, A. 1972, Isaac Newton’s PhilosophiaeNaturalis Principia Mathematica, Facsimile of third edition (1726) with vari-ant readings, 2 vols.) (Cambridge, MA: Harvard University Press).

Copernicus. 1938–1941. de Revolutionibus Preface and Part of Vol. 1 plus notes.Occas. Notes R. Astr. Soc. 1, 1

Crombie, A. C. 1959. Medieval and Early Modern Science (2 vols.) (Garden City,NY: Anchor Books).

. 1971. Robert Grossteste and the Founding of Experimental Science (Oxford:Oxford University Press).

Crossby, H. L. (ed.) 1961. Thomas of Bradwardine—His Tractatus de Proportion-ibus (Madison: University of Wisconsin Press).

Darrigol, O. 2000. Electrodynamics from Ampere to Einstein (Oxford: Oxford Uni-versity Press).

Dijsterhuis, E. 1987. Archimedes (Princeton, NJ: Princeton University Press).Dirac, P. A. M. 1965. The Principles of Quantum Mechanics (3rd ed.) (Oxford:

Oxford University Press).. 1978. Directions in Physics (New York: Wiley).

Dobbs, B. J. T. 1975. The Foundations of Newton’s Alchemy or ‘The Hunting ofthe Greene Lyon’ (Cambridge, UK: Cambridge University Press).

Drabkin, I. E. and Drake, S. 1960. Galileo on Motion (Madison: University ofWisconsin Press).

Drachmann, A. G. 1963. The Mechanical Technology of the Greeks and Romans(Madison: University of Wisconsin Press).

Drake, S. (ed.) 1953. Galileo Galilei: Dialogue Concerning the Two Chief WorldSystems (Berkeley: University of California Press).

Drake, S. 1957. Discoveries and Opinions of Galileo (New York: Anchor).Drake, S. (trans, ed.) 1974. Galileo: Two New Sciences (Madison: University of

Wisconsin Press).Drake, S. 1978. Galileo at Work: His Scientific Biography (Chicago: University of

Chicago Press).Drake, S. 1981. Cause, Experiment, and Science: A Galilean Dialogue, Incorpo-

rating a New English Translation of Galileo’s Bodies That Stay Atop Water,or Move in It Cause, Experiment, and Science (Chicago: University of ChicagoPress).

Bibliography 223

Drake, S. and Drabkin, I. E. 1969. Mechanics in Sixteenth-Century Italy: Selectionsfrom Tartaglia, Benedetti, Guido Ubaldo, and Galileo (Madison: University ofWisconsin Press).

Dreyer, J. L. E. 1953. A History of Astronomy from Thales to Kepler (New York:Dover).

Effler, R. R. 1962. John Duns Scotus and the Principle of “Omni Quod Movetur abalio Movetur” (New York: Fraciscan Inst. Publ.).

Einstein, A. 1990. Collected Papers of Albert Einstein (vols. 2 and 6) (Princeton,NJ: Princeton University Press).

Elkana, Y. 1974. The Discovery of the Conservation of Energy (Cambridge, MA:Harvard University Press).

Freeman, K. 1977. Ancilla to the Pre-Socratic Philosophers (Cambridge, MA:Harvard University Press).

Gaukroger, S. 1995. Descartes: An Intellectual Biography (Oxford: Oxford Univer-sity Press).

Gibbs, J. W. 1949. The Collected Papers of J. Willard Gibbs (vol. 1) (New Haven,CT: Yale University Press).

Gilbert, W. 1958. On the Magnet: de Magnete, a Translation by the Gilbert Society,1900 (New York: Basic Books).

Gillespie, H. 1960. The Edge of Certainty: An Essay in the History of ScientificIdeas (Princeton, NJ: Princeton University Press).

Gilmore, C. S. 1971. Coulomb and the Evolution of Physics and Engineer-ing in Eighteenth-Century France (Princeton, NJ: Princeton UniversityPress).

Goddu, A. 2001. “The Impact of Ockham’s Reading of the Physics on the Merto-nians and Parisian Terminists.” Early Sci. Med., 6, 204.

Goldstein, H. H. 1980. A History of the Calculus of Variations from the 17thCentury through the 19th Century (Berlin: Springer-Verlag).

Gowing, R. 1983. Roger Cotes: Natural Philosopher (Cambridge UniversityPress).

Grant, E. 1977. Physical Science in the Middle Ages (New York: Wiley).. 1981. Much Ado About Nothing: Theories of Space and Vacuum from the

Middle Ages to the Scientific Revolution (Cambridge, UK: Cambridge Univer-sity Press).

. 1996. Planets, Stars, and Orbs: The Medieval Cosmos 1200–1687 (Cam-bridge, UK: Cambridge University Press).

Gratton-Guiness, I. 1995. “Why Did George Green Write His Essay of 1828 onElectricity and Magnetism?” Amer. Math. Monthly, 102, 387.

Green, G. 1871 (rpt. 1970). The Mathematical Papers of George Green (New York:Chelsea).

Hall, A. R. 1952. Ballistics in the Seventeenth Century (Cambridge, UK: CambridgeUniversity Press).

Hall, A. R. and Hall, M. B. 1962. Unpublished Scientific Papers of Isaac Newton(Cambridge, UK: Cambridge University Press).

Hamilton, W. R. 1931. The Mathematical Papers of William Rowan Hamilton,Vol. 2: Dynamics and Vol. 3: Algebra (Cambridge, UK: Cambridge UniversityPress).

224 Bibliography

Hardie, R. P., Gaye, R. K., and Stocks, J. L. (trans.) 1930. The Works of Aristotle:Physica, De Caelo (Oxford: Oxford University Press).

Harrison, E. 2000. Cosmology, The Science of the Universe (2nd ed.) (Cambridge,UK: Cambridge University Press).

Hart, I. B. 1963. The Mechanical Investigations of Leonardo da Vinci (Berkeley:University of California Press).

Heath, T. L. 1897. The Works of Archimedes (New York: Dover).Heilbron, J. L. 1979. Electricity in the 17th and 18th Centuries: A Study of Early

Modern Physics (Berkeley: University of California Press).Herivel, J. L. 1965. The Background to Newton’s Principia: A Study of New-

ton’s Dynamical Researches in the Years 1664–84 (Oxford: Oxford UniversityPress).

Hertz, H. 1956. The Principles of Mechanics (version of 1899) (New York: Dover).Hesse, M. 1961. Forces and Fields (New York: Dover).Holton, E. 1988. Thematic Origins of Scientific Thought from Kepler to Einstein

(Cambridge, MA: Harvard University Press).Jaki, S. L. 1969. Paradox of Olbers Paradox: A Case History of Scientific Thought

(London: Herder).Jammer, M. 1999. Concepts of Force: A Study in the Foundations of Dynamics

(New York: Dover).Jardine, N. 1982. “The Significance of the Copernican Orbs.” JHA, 13, 168.Joule, J. P. 1850. “On the Mechanical Equivalent of Heat.” Phil Trans. R. Soc.

140, 61.Jourdain, P. E. B. (ed.) 1908. Abhandlungen uber die Prizipiender Mechanik von

Labgrange, Rodrigues, Jacobi, und Gauss (Ostwald’sKlassiker der ExactenWissenschaften Nr. 167 (Leipzig: Wilhelm Engelmann).

. 1917. Supplement to Ernst Mach’s “The Science of Mechanics 3rd Engl.Ed.” (Chicago: Open Court).

Jungnickel, C. and McCormack, R. 1986. The Intellectual Mastery of Nature:Theoretical Physics from Ohm to Einstein, Vol. 2: The Now Mighty TheoreticalPhysics 1970–1925 (Chicago: University of Chicago Press).

. 1999. Cavendish: An Experimental Life (Lewisburg, PA: Bucknell Uni-versity Press).

Kirschner, S. 2000. “Oresme on Intension and Remission of Qualities in HisCommetary on Aristotle’s Physics.” Vivarium, 38, 255.

Koertge, N. 1977. “Galileo and the Problem of Accidents.” J. Hist. Ideas, 38, 389.Koyre, A. 1955. “A Documentary History of the Problem of Fall from Kepler to

Newton.” Trans. Amer. Phil. Soc., 45(4), 329.. 1956. From the Closed World to the Infinite Universe (Baltimore, MD:

Johns Hopkins University Press).. 1967. Newtonian Studies (Cambridge, MA: Harvard University Press).. 1973. The Astronomical Revolution: Copernicus, Kepler, and Galileo

(Ithaca, NY: Cornell University Press).Kragh, H. and Carazza, B. 2000. “Classical Behavior of Macroscopic Bodies from

Quantum Principles: Early Discussions.” Arch. Hist. Exact Sci., 55, 43.Kuhn, T. S. 1957. The Copernican Revolution: Planetary Theory in the Development

of Western Thought (Cambridge, MA: Harvard University Press).

Bibliography 225

Lanczos, C. 1970. The Variational Principles of Mechanics (Toronto: University ofToronto Press).

Laplace, P. S. 1965. Mecanique Celeste (trans. N. Bowditch, 5 vols.) (New York:Chelsea).

Lorentz, H. A., Einstein, A., Minkowski, H., and Sommerfeld, A. 1923. ThePrinciple of Relativity (New York: Dover).

MacCurdy, E. 1939. The Notebooks of Leonardo da Vinci (New York: GeorgeBaziller).

Mach, E. 1907. The Science of Mechanics (7th ed.) (Chicago: Open Court).Maier, A. 1982. On the Threshold of Exact Science: Selected Writings of Anneliese

Maier on Late Medieval Natural Philosophy (ed. S. D. Sargent) (Philadelphia:University of Pennsylvania Press).

Maxwell, J. Clerk. 1920. Matter and Motion (Cambridge, UK: Cambridge Univer-sity Press; rpt. 1991, New York: Dover).

. 1965. The Scientific Papers of James Clerk Maxwell (ed. E. Nevins) (Rpt.,New York: Dover).

McMullin, E. (ed.) 1968. Galileo: Man of Science (New York: Basic Books).Meli, D. B. 1993. Equivalence and Priority: Newton versus Leibniz (Oxford: Oxford

University Press).Mendoza, E. (ed.) 1960. Reflections on the Motive Power of Heat: Sadi Carnot, E.

Clapeyron, and R. Clausius (New York: Dover).Miller, A. I. 1986. Frontiers of Physics: 1900–1911, Selected Essays (Zurich:

Birkhauser).. 1998. Albert Einstein’s Special Theory of Relativity: Emergence (1905)

and Early Interpretation (1905–1911) (New York: Springer-Verlag).Moody, E. A. 1951. “Galileo and Avempace: The Dynamics of the Leaning Tower

Experiment: I, II.” J. Hist. Ideas, 12, 163, 375.Moody, E. and Clagett, M. 1952. The Medieval Science of Weights (Wisconsin:

University of Wisconsin Press).Needham, J. 1962. Science and Civilization in China, Vol. IVA: Physics (Cambridge,

UK: Cambridge University Press).Ottman, J. and Wood, R. 1999. “Walter of Burley: His Life and Works.” Vivarium,

37, 1.Pedersen, O. 1981. “The Origins of the ‘Theorica Planetarum.’” JHA, 12, 113.Pedersen, O. and Pihl, M. 1972. Early Physics and Astronomy (New York: Science

History Publ.).Petoski, H. 1994. Design Paradigms: Case Histories of Error and Judgment in

Engineering (Cambridge, UK: Cambridge University Press).Piaget, J. and Inhelder, B. 1958. The Growth of Logical Thinking from Childhood

to Adolescence (New York: Basic Books).Pines, S. (ed.) 1963. Moses Maimonides: The Guide of the Perplexed (2 vols.)

(Chicago: University of Chicago Press).Pugliese, P. J. 1989. “Robert Hooke and the Dynamics of a Curved Path” in M.

Hunter and S. Schaffer (eds.), Robert Hooke: New Studies (London: BoydenPress), p. 181.

Reti, L. (ed.) 1974. The Unknown Leonardo (New York: McGraw-Hill).Rouse, H. and Ince, S. 1957. History of Hydraulics (New York: Dover).

226 Bibliography

Rumford (Benjamin Count of Rumford) 1789. “An Inquiry Concerning the Sourceof the Heat Which Is Excited by Friction.” Phil Trans. R. Soc., 88, 80.

Sarton, G. An Introduction to the History of Science (5 vols.) (Baltimore, MD:Williams and Wilkins).

Schwinger, J. ed. 1958. Selected Papers on Quantum Electrodynamics (New York:Dover).

Shapin, S. 1996. The Scientific Revolution (Chicago: University of Chicago Press).Sommerfeld, A. 1943. Mechanics (New York: Academic Press).Southern, R. 1986. Robert Grossteste: The Growth of an English Mind in Medieval

Europe (Oxford: Oxford University Press).Spires, I. H. B., Spures, A. G. H., and Barry, F. (eds.) 1937. The Physical Treatises

of Pascal (New York: Columbia University Press).Stephenson, B. 1987. Kepler’s Physical Astronomy (Princeton, NJ: Princeton Uni-

versity Press).Thijssen, J. M. M. H. 2004.“The Burdian School Reassessed: John Buridan and

Albert of Saxony.” Vivarium, 42, 18.Thomson, W. (Lord Kelvin) 1882. The Mathematical and Physical Papers of Lord

Kelvin (5 vols.) (Cambridge University Press).Thomson, W. and Tait, P. G. 1912. Elements of Natural Philosophy (2 vols.) (New

York: Dover).Thorndike, L. 1923–1958. A History of Magic and Experimental Science (8 vols.)

(New York: Columbia University Press).Todhunter, I. 1873 (rpt. 1962). A History of the Mathematical Theories of Attraction

and the Figure of the Earth (from the Time of Newton to That of Laplace) (NewYork: Dover).

Tricker, R. A. R. 1965. Early Electrodynamics: The First Law of Circulation(Oxford: Pergamon).

von Helmholtz, Hermann. 1971. “The Conservation of Force: A Physical Mem-oire (1847),” in R. Kahl (ed.), Selected Writings of Hermann von Helmholtz(Middletown, CT: Wesleyan University Press), p. 2.

Wallace, W. A. 1978. “Causes and Forces in Sixteenth-Century Physics.” Isis,69, 400.

Weisheipl, J. A. 1959. “The Place of John Dumbleton in the Merton School.” Isis,50, 439.

. 1963. “The Principle Omnequod Movetur ab alio movetur in MedievalPhysics.” Isis, 56, 26.

. 1971. The Development of Physical Theory in the Middle Ages (Ann Arbor:University of Michigan Press).

Westfall, R. S. 1971. Force in Newton’s Physics (New York: Neale Watson).. 1971. The Construction of Modern Science: Mechanisms and Mechanics

(New York: Wiley).. 1982. Never at Rest: A Biography of Issac Newton (Cambridge, MA:

Cambridge University Press).Whiteside, D. T. 1970. “Before the Principia: The Maturing of Newton’s Thoughts

on Dynamical Astronomy 1664–1684.” JHA, 1, 5.. 1976. “Newton’s Lunar Theory: From High Hope to Disenchantment.”

Vistas in Astr., 19, 317.

Bibliography 227

Williams, L. P. 1971. The Selected Correspondence of Michael Faraday (2 vols.)(Cambridge, UK: Cambridge University Press).

Wilson, C. 1972. “How Did Kepler Discover His First Two Laws?” Sci. Amer.,226(3), 93.

. 1974. “Newton and Some Philosophers on Kepler’s Laws.” J. Hist. Ideas,35, 231.

Woodruff, A. E. 1962. “Action at a Distance in Nineteenth Century Electrody-namics.” Isis, 53, 439.

Woodward, R. S. 1895. “An Historical Survey of the Science of Mechanics.”Science, 1, 141.


INDEX

Accelerated motion (Merton), difform difform,31–3, 48, 55. See Violent motion

Acoustics, and electromagnetism, 154. See alsoVibration

Action, 86–7; and vis viva, 85–87, 109–12.See also Least action, principle of

Action-at-a-distance, 141–3Ampere, Andre-Marie, 140–1Anaxagoras, 3Anaximander, 3Ancient physics, cultural setting, 3–4Archimedes, 9–12, 36n1, 40Architecture. See StaticsAristarchus of Samos, 40Aristotle, 4–7; causes of motion. See Causes,

AristotelianAstrology, 20n4Atmosphere, 58–6, 98n11, 131–3Atom, ancient conception of, 5, 18; Newtonian,

100–2, 120, 126, 128–9; vortex, 179–80. Seealso Bohr, Niels; quantum mechanics

Atomic spectra, 180–1, 183–4, 185, 200n5

Barometer, 56Bernoulli, Daniel, 93; and gas laws, 100–1; law

for fluid motion, 93–4Biomechanics, 62n2; and Galileo, 50, 52,

62n5–6; and Leonardo, 35–6Blackbody radiation, 182–3Black hole. See Schwarzschild metric

Bohr atom. See Bohr, NielsBohr, Niels, atomic model, 185–6;

correspondence principle, 188;complementarity principle, 193

Boltzmann, Ludwig, 126, 154, 182Born, Max, 191–3Boscovich, Roger, 103, 120Boyle, Robert, 57, 100–1Bradwardine, Thomas, 28–31Brahe, Tycho. See Tycho BraheBrownian motion, 128–9Buridan, Jean, 27–8, 54

Carnot, Sadi, 122–5; Carnot cycle, 124–5Cartesian mechanics, 58–61, 75–7Causes, Aristotelian (material, formal, efficient,

final), 4–5, 19n1Cavendish Henry, 107Celestial mechanics, 81–5Celestial motions, Aristotelian. See EudoxusCenter of gravity, 12Centrifugal force, 59–60, 69–71, 84, 92n4, 95,

97n4, 121Centripetal force, 68–71. See also GravitationChandler wobble, 97Chaos, 130–1Chladni, E. F. F., 91Circuits, as flows, 143–5. See Electrical circuitsClapyron, Benoit, 123Clausius, Rudolph, 123, 126–7

230 Index

Compton wavelength, 189Condemnations of 1210 and 1277, 23Conjunctus, 60. See also Centrifugal forceCopenhagen interpretation of quantum

mechanics, 193Copernicus, Nicholas, 40–3Coriolis force, 95–7, 132Coriolis, Gaspard Gustav de, 96Coulomb, Charles Augustin, studies of friction

92–3, 107, 138–9, 142Curvature, and gravitation, 166–9, as tidal

acceleration, 175

D’Alembert, Jean le Rond, 109, 111Dark matter, 176–7Dark night sky paradox, 98n8De Broglie, Louis, 189–90De Broglie wavelength, 190Descartes, Rene, 58–61Difform (uniform and difform). See Accelerated

motionDirac, Paul A. M., 194–6Displacement (electrical), 149–50Dynamics, and electron theory, 155–6. See also

Lorentz force

Edmund Halley, and tides, 78–9Einstein, Albert, and Brownian motion, 128–9;

and photoelectric effect, 183–4; andRelativity theory, 159–177

Elasticity, 87–92. See also Statics, beams;Statics, striings

Electrical charge, 141–2; and electron, 180,181; units of, 148

Electrical circuits, 141, 142–5Electrical current, 141, 150, 157n5Electrical fluid, 138–40, 141–2Electric field, 149–51Electrodynamics. See Ampere, Andre-Marie;

Electromagnetism, dynamical fieldElectromagnetic units, and dynamics,

148Electromagnetism, 140–1; dynamical field,

149–51; energy of, 151; flux of, 154;pressure, 154; waves, 153,

Electromotive force (EMF), 143–4, 147–8.See also Voltic pile

Electrostatic potential. See Potential field;Voltic pile

Electrostatics, 137–9, 142–3; units of, 148

Energy, conservation of, 144–5, 156n2; andvirial theorem, 126–7

Entropy, 122, 124. See also ThermodynamicsEpicycles. See Ptolemy, ClaudiusEpicyclic motion, 16–17, 41–2Equilibrium, 10–11, 25–6Equivalence principle, and General relativity,

164–5Ether, Cartesian, 60–1; Maxwell, 151–3Eudoxus, planetary motion, 8–9, 13–17Euler, Leonard, 84, 87–92, 110–11

Faraday, Michael, 145–8Fermi, Enrico, 199Feynman, Richard, 196–8Field as force, 102–3Fluids, 75–6, 111, 133, 152. See also ViscosityFontenelle, Conversations on the Plurality of

Worlds, 75Forces, law of addition of, 67–8; in special

relativity, 163Foucault, J. B. Leon, 96–7Frames of reference, noninertial, 95–7Franklin, Benjamin, 138–9Freefall, 15, 49–50, 97n2. See also General

theory of relativity, equivalence principleFriction, 92–3

Galilean invariance, 54. See also Special theoryof relativity

Galileo Galilei, 47–56; De Motu (On Motion),49; Dialogs Concerning the Two Chief WorldSystems, 54; Letters on Sunspots, 54;Sidereus Nuncius (Starry Messenger), 52–3;Two New Sciences, 51–2

Gassendi, Pierre, 57Gauss, Karl Friedrich, 105; theorem for fields,

114n2General Scholium. See Newton, General

Scholium, PrincipiaGeneral theory of relativity, equivalence

principle, 164–5Geodesy and the measurement of the density of

the Earth, 107–6Geostrophic motion. See AtmosphereGibbs, Josiah Willard, 126Gilbert, William, 43–4, 140Gouvernor, 121Gravitas (weight), 5Gravitational lens, 173–4, 177n3

Index 231

Gravitational waves, 174–5Gravitation, Newtonian force law, 107; and

geometry, 165–70; andGeneral relativity, 164–176Green, George, 103–9, 197Grossmann, Marcel, 167Guericke, Otto von, 56–7Gyroscope, 96–7

Halley, Edmund, 66, 98n8; comet, 83–4Hamilton, William Rowan, 111–12, 127Harmonic oscillator, 88; quantized, 187, 192,

196Heat, and work, 120–1; as a fluid, 123–5. See

also EntropyHeat capacity, 125, 135n4Heisenberg, uncertainty principle, 192–3Heisenberg, Werner, 191–6Heliocentric universe. See Copernicus,

NicholasHelmholtz, Hermann von, 93, 133, 134, 144–5,

152, 156, 190Hero of Alexandria, 12, 13, 121Hertz, Heinrich, 153Homeoids, 71, 73–4Hooke, Robert, 61–2; Hooke constant. See

Elasticity; Statics, beams; Statics, springsHuygens, Christian, 59–60Hydrostatics, Archimedes, 11–13; Chinese,

19n3

Impetus, 27–8. See also Mertonian mechanicsInduction and lines of force, 145–7Industrial revolution, 118–9Inertia, 55, 58–60, 71–2, 160–1, 164Islamic physics and astronomy, 18–19, 23,

29

Jacobi, Karl Gustav Jacob, 111Jordanus de Nemore, 24Joule, James Prescott, 125Jupiter, moons of, 52–3, 63n7

Kelvin, Lord (William Thomson), 107–8, 133,134, 152, 179

Kepler, Johannes, 45–7; laws of planetarymotion, 46–7

Kepler problem, 65–6, 81. See also Two-bodyproblem

Kirchhoff, Gustav Robert, 144

Lagrange, Joseph-Louis, 90, 110, 127–8.See also Three-body problem

Lagrangian points, 84. See also Three-bodyproblem

Langevin, Paul, 128–9, 190Laplace, Pierre-Simone, 84, 103–5, 120–1;

collaboration with Lavoisier, 120–1Laws of motion, Cartesian, 58–9; Mertonian (see

Mertonian mechanics); Newtonian, 67–8Least action principle, 109–12; and quantum

mechanics, 196–8Leibniz, Gottfried, 85–7Leonardo da Vinci, 35–6, 62n6Lever, 10–14, 25Leyden jar (condenser, capacitor), 142Light, constancy of speed of, 163, 177n1;

gravitational lens, 173–4; momentum of,173, 189

Light. See Electromagnetism, dynamical fieldLines of force. See Magnetic fieldLorentz factor, 163Lorentz force, 156Lorentz, Henrik Anton, 156Lucretius, 18. See also Atom

Mach, Ernst, 96; Mach’s principle, 96. See alsoInertia

Machines, and thermal cycles, 121; types of.See Hero of Alexandria

Maclaurin, Colin, 67, 84Magnetic field, 145, 147; and atoms, 180–1;

dynamo, 147–8; lines of force, 145–6;transformer, 147; units of, 148, 149

Magnetic induction, 145–6, 155Magnetism, 43–4, 47, 149Magnus force, 179–80Maimonedes, Moses, 21Maskelyne, Nevil, 106–7Matrix mechanics and quantum mechanics,

191–6. See also Heisenberg, Werner; Born,Max

Maxwellian velocity distribution function,127–8

Maxwell, James Clerk, 126, 149–53,Mechanical equivalent of heat, 125Medieval physics, cultural setting of, 22–24,

37n3Medieval statics, 24–26Merton College, Oxford. See Mertonian

mechanics

232 Index

Mertonian mechanics, 26, 28–33, 47–8, 55;Merton rule (mean value theorem), 30,32

Metric, 168, 177n3Microscopic forces, 198–200Millikan, Robert, 181Models, xiv, 17Motion, Aristotelian classification, 5–8. See

also Laws of motion

Natural motion, 5Natural place, 5, 15Neptune, discovery of, 85Newton, General Scholium, Principia, 79–81Newton, Isaac, 65ff; early work on lunar

motion, 65–6; Laws of motion, 67–8; Optiks,queries of, 101–2; Principia,66ff

Nicholas of Cusa, 54Normal force, 92–2Nuclear atom. See AtomNuclear force. See Microscopic forces

Oersted, Hans Christian, 139–40Ohm, Georg Simon, 143–4Orbital motion, 72; in General relativity, 170–2;

perturbations, 81–5. See also Kepler problemOresme, Nicole, 31–3

Paris. See also Condemnations of 1210 and1277; Parisian mechanics; Universties,medieval

Parisian mechanics, 26–8, 31–3Parmenides, 7Pascale, Blaise, 57–8Pauli exclusion principle, 186Pendulum, 50, 62n4, 76–7, 97n6; Foucault,

96Perpetual motion, 119Phase space, 127Photoelectric effect, 183–4Photon, 183Physical units (dimensions), 62n3Pisa, 48Planck, Max, 182–3; Planck’s constant 183.

See also Blackbody radiationPlanetary motions, 8f. See also EpicyclesPlanetary vortices, 60–1,75, 99–100Plato, 3–4Poincare, Henri, 130–1

Poisson, Simeon-Denis, 105; Poisson bracket,195, 201n7. See also Laplace, Pierre-Simone

Polarization, 149–50Positional gravity, 24–5, 62n4Potential field, 103–5, 107–8, 139, 167,

169–70, 173Potential, scalar. See Potential fieldPotential, vector, 150, 194Power. See WorkPoynting, John Henry, 154Precession, 97; orbital precession in General

relativity, 170–1Pressure, gas 55–8, 132; radiation, 153–5,

157n6Priestly, Joseph, 142Principia Mathematicae Naturalis. See Newton,

IsaacPrime mover, 7, 9, 16, 41–2Probability, 135n3Propagator, 196–7. See also Least action

principle, and quantum mechanicsPtolemy, Claudius, 15–17; Almagest, 16;

Tetrabiblos, 17

Radiation pressure, 153–5, 157n6Redshift, gravitational, 169, 192Relativity of motion, Galilean invariance, 54,

162; Lorentz invariance, 162Relativity of motion, General relativity,

114n3Renaissance physics, cultural setting of,

33–35Rest mass, 163–4Restricted theory of relativity. See Special

theory of relativityRetrograde motion, 8. See also EpicyclesReynolds number, 94Riemann, Bernard, 168Rotational motion and frames of reference,

95–7Rutherford, Ernest, 184–5

Scholastic dynamics, 26–35Scholastics (Schoolmen). See Universities,

medievalSchrodinger, Erwin, 190–1; Heisenberg,

Werner, 191–6Schwarzschild, Karl, 171–2; Schwarzschild

metric, 171–2Seismology. See Vibration

Index 233

Self-motion, 7Shear, 90Spacetime. See Special theory of relativity;

General theory of relativitySpecial theory of relativity, 160–4; and

electromagnetism, 161–2; equivalence ofmass and energy, 164; velocity additionformula, 163

Spin, electron, 186–7Stars, as self-gravitating compressible masses,

133–5States of matter, 126; of motion, 6Statics, Archimedes, 9–13, 40; architecture,

34–5; Galilean, 50–2; medieval, 24–6Statics, beams, 35, 37n4, 51–2, 62–3; Euler’s

law for beams, 89–90Statics, springs, 62; Hooke’s law, 89–90Statistical mechanics, 125–135; and quantum

theory, 182–4Steam engine, 119, 121, 122; Newcomen

engine, 121. See also Carnot, SadiStokes, George G. 94Stress, 61. See also Statics, beamsSum over paths. See Least action principle,

quantum mechanics

Thermal cycle 122–3; efficiency, 125Thermodynamics, 117–126; first law of, 124,

145. See also Entropy; HeatThomson, John Joseph, 180, 181, 185Three-body problem, 84–5, 130–1Tides, 54–5, 72–3, 78–9; and General theory of

relativity, 175Time, 5. See also Relativity of motionTorricelli, Evangelista, 56Torsion, 90Transition probabilities, 183–4

Two-body problem, in General theory ofrelativity, 170–1. See also Kepler problem

Tycho Brahe, 44–5

Uncertainty principle. See Heisenberg,uncertainty principle

Units. See Physical units (dimensions)Universties, medieval, 23–4, 26

Vacuum (void), force of, 56–7Vibration: of beams, 89–90; of plates, 91–2;

seismology, Chang Heng and inventionof seismograph, 92; of strings, 89

Violent motion, 6; and impetus, 27–9Virial theorem, 126–7, 134Viscosity, 93–4Vis mortis. See WeightVis viva (Kinetic energy), 86–7. See also

Helmholtz, Hermann vonVitruvius, 12Voltic pile (electrical battery), 140,Vortex motion, 60–1, 133, 152, 179–80;

atmosphere, 96, 98n11

Watt, James, 121Wave mechanics, 189–91. See also de Broglie,

Louis; Schrodinger, ErwinWeight, 24, gravitas, 5, 10f. See also Normal

forceWilliam of Morbeke, 22Work, medieval notion, 25; thermodynamics,

122–4

Zeeman, Peter, 180–1Zeno of Elea, paradoxes of motion, 6Zero point energy, of quantized harmonic

oscillator, 196


About the Author

STEVEN N. SHORE (PhD, Toronto 1978) is a professor of physics in the Depart-ment of Physics “Enrico Fermi” of the Univesity of Pisa and is associated withIstituto Nazionale di Astrofisica (INAF) and Istituto Nazionale di Fisica Nucleare(INFN) before which he was chair of the Department of Physics and Astronomyat Indiana University South Bend. His previous books include The Encyclopediaof Astronomy and Astrophysics (Academic Press, 1989), The Tapestry of ModernAstrophysics (Wiley, 2003), and Astrophysical Hydrodynamics: An Introduction(Wiley/VCH, 2007). He is an associate editor of Astronomy and Astrophysics.

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